hypothesis testing in r cheat sheet

Statistics Made Easy

The Complete Guide: Hypothesis Testing in R

A hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis.

This tutorial explains how to perform the following hypothesis tests in R:

One sample t-test
Two sample t-test
Paired samples t-test

We can use the t.test() function in R to perform each type of test:

x, y: The two samples of data.
alternative: The alternative hypothesis of the test.
mu: The true value of the mean.
paired: Whether to perform a paired t-test or not.
var.equal: Whether to assume the variances are equal between the samples.
conf.level: The confidence level to use.

The following examples show how to use this function in practice.

Example 1: One Sample t-test in R

A one sample t-test is used to test whether or not the mean of a population is equal to some value.

For example, suppose we want to know whether or not the mean weight of a certain species of some turtle is equal to 310 pounds. We go out and collect a simple random sample of turtles with the following weights:

Weights : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

The following code shows how to perform this one sample t-test in R:

From the output we can see:

t-test statistic: -1.5848
degrees of freedom: 12
p-value: 0.139
95% confidence interval for true mean: [303.4236, 311.0379]
mean of turtle weights: 307.230

Since the p-value of the test (0.139) is not less than .05, we fail to reject the null hypothesis.

This means we do not have sufficient evidence to say that the mean weight of this species of turtle is different from 310 pounds.

Example 2: Two Sample t-test in R

A two sample t-test is used to test whether or not the means of two populations are equal.

For example, suppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, we collect a simple random sample of turtles from each species with the following weights:

Sample 1 : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

Sample 2 : 335, 329, 322, 321, 324, 319, 304, 308, 305, 311, 307, 300, 305

The following code shows how to perform this two sample t-test in R:

t-test statistic: -2.1009
degrees of freedom: 19.112
p-value: 0.04914
95% confidence interval for true mean difference: [-14.74, -0.03]
mean of sample 1 weights: 307.2308
mean of sample 2 weights: 314.6154

Since the p-value of the test (0.04914) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean weight between the two species is not equal.

Example 3: Paired Samples t-test in R

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump (in inches) of basketball players.

To test this, we may recruit a simple random sample of 12 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month.

The following data shows the max jump height (in inches) before and after using the training program for each player:

Before : 22, 24, 20, 19, 19, 20, 22, 25, 24, 23, 22, 21

After : 23, 25, 20, 24, 18, 22, 23, 28, 24, 25, 24, 20

The following code shows how to perform this paired samples t-test in R:

t-test statistic: -2.5289
degrees of freedom: 11
p-value: 0.02803
95% confidence interval for true mean difference: [-2.34, -0.16]
mean difference between before and after: -1.25

Since the p-value of the test (0.02803) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean jump height before and after using the training program is not equal.

Additional Resources

Use the following online calculators to automatically perform various t-tests:

One Sample t-test Calculator Two Sample t-test Calculator Paired Samples t-test Calculator

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Learn R: Hypothesis Testing

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Sample Vs. Population Mean

In statistics, we often use the mean of a sample to estimate or infer the mean of the broader population from which the sample was taken. In other words, the sample mean is an estimation of the population mean .

Hypothesis Test P-value

Statistical hypothesis tests return a p-value , which indicates the probability that the null hypothesis of a test is true. If the p-value is less than or equal to the significance level , then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the p-value is greater than the significance level, then the null hypothesis is not rejected.

Hypothesis Test Errors

Type I errors, also known as false positives , is the error of rejecting a null hypothesis when it is actually true. This can be viewed as a miss being registered as a hit. The acceptable rate of this type of error is called significance level and is usually set to be 0.05 (5%) or 0.01 (1%).

Type II errors, also known as false negatives , is the error of not rejecting a null hypothesis when the alternative hypothesis is the true. This can be viewed as a hit being registered as a miss.

Depending on the purpose of testing, testers decide which type of error to be concerned. But, usually type I error is more important than type II .

Central Limit Theorem

The central limit theorem states that as samples of larger size are collected from a population, the distribution of sample means approaches a normal distribution with the same mean as the population. No matter the distribution of the population (uniform, binomial, etc), the sampling distribution of the mean will approximate a normal distribution and its mean is the same as the population mean.

The central limit theorem allows us to perform tests, make inferences, and solve problems using the normal distribution, even when the population is not normally distributed.

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Hypothesis Tests in R

This tutorial covers basic hypothesis testing in R.

Normality tests
Shapiro-Wilk normality test
Kolmogorov-Smirnov test
Comparing central tendencies: Tests with continuous / discrete data
One-sample t-test : Normally-distributed sample vs. expected mean
Two-sample t-test : Two normally-distributed samples
Wilcoxen rank sum : Two non-normally-distributed samples
Weighted two-sample t-test : Two continuous samples with weights
Comparing proportions: Tests with categorical data
Chi-squared goodness of fit test : Sampled frequencies of categorical values vs. expected frequencies
Chi-squared independence test : Two sampled frequencies of categorical values
Weighted chi-squared independence test : Two weighted sampled frequencies of categorical values
Comparing multiple groups: Tests with categorical and continuous / discrete data
Analysis of Variation (ANOVA) : Normally-distributed samples in groups defined by categorical variable(s)
Kruskal-Wallace One-Way Analysis of Variance : Nonparametric test of the significance of differences between two or more groups

Hypothesis Testing

Science is "knowledge or a system of knowledge covering general truths or the operation of general laws especially as obtained and tested through scientific method" (Merriam-Webster 2022) .

The idealized world of the scientific method is question-driven , with the collection and analysis of data determined by the formulation of research questions and the testing of hypotheses. Hypotheses are tentative assumptions about what the answers to your research questions may be.

Formulate questions: How can I understand some phenomenon?
Literature review: What does existing research say about my questions?
Formulate hypotheses: What do I think the answers to my questions will be?
Collect data: What data can I gather to test my hypothesis?
Test hypotheses: Does the data support my hypothesis?
Communicate results: Who else needs to know about this?
Formulate questions: Frame missing knowledge about a phenomenon as research question(s).
Literature review: A literature review is an investigation of what existing research says about the phenomenon you are studying. A thorough literature review is essential to identify gaps in existing knowledge you can fill, and to avoid unnecessarily duplicating existing research.
Formulate hypotheses: Develop possible answers to your research questions.
Collect data: Acquire data that supports or refutes the hypothesis.
Test hypotheses: Run tools to determine if the data corroborates the hypothesis.
Communicate results: Share your findings with the broader community that might find them useful.

While the process of knowledge production is, in practice, often more iterative than this waterfall model, the testing of hypotheses is usually a fundamental element of scientific endeavors involving quantitative data.

The Problem of Induction

The scientific method looks to the past or present to build a model that can be used to infer what will happen in the future. General knowledge asserts that given a particular set of conditions, a particular outcome will or is likely to occur.

The problem of induction is that we cannot be 100% certain that what we are assuming is a general principle is not, in fact, specific to the particular set of conditions when we made our empirical observations. We cannot prove that that such principles will hold true under future conditions or different locations that we have not yet experienced (Vickers 2014) .

The problem of induction is often associated with the 18th-century British philosopher David Hume . This problem is especially vexing in the study of human beings, where behaviors are a function of complex social interactions that vary over both space and time.

Falsification

One way of addressing the problem of induction was proposed by the 20th-century Viennese philosopher Karl Popper .

Rather than try to prove a hypothesis is true, which we cannot do because we cannot know all possible situations that will arise in the future, we should instead concentrate on falsification , where we try to find situations where a hypothesis is false. While you cannot prove your hypothesis will always be true, you only need to find one situation where the hypothesis is false to demonstrate that the hypothesis can be false (Popper 1962) .

If a hypothesis is not demonstrated to be false by a particular test, we have corroborated that hypothesis. While corroboration does not "prove" anything with 100% certainty, by subjecting a hypothesis to multiple tests that fail to demonstrate that it is false, we can have increasing confidence that our hypothesis reflects reality.

Null and Alternative Hypotheses

In scientific inquiry, we are often concerned with whether a factor we are considering (such as taking a specific drug) results in a specific effect (such as reduced recovery time).

To evaluate whether a factor results in an effect, we will perform an experiment and / or gather data. For example, in a clinical drug trial, half of the test subjects will be given the drug, and half will be given a placebo (something that appears to be the drug but is actually a neutral substance).

Because the data we gather will usually only be a portion (sample) of total possible people or places that could be affected (population), there is a possibility that the sample is unrepresentative of the population. We use a statistical test that considers that uncertainty when assessing whether an effect is associated with a factor.

Statistical testing begins with an alternative hypothesis (H 1 ) that states that the factor we are considering results in a particular effect. The alternative hypothesis is based on the research question and the type of statistical test being used.
Because of the problem of induction , we cannot prove our alternative hypothesis. However, under the concept of falsification , we can evaluate the data to see if there is a significant probability that our data falsifies our alternative hypothesis (Wilkinson 2012) .
The null hypothesis (H 0 ) states that the factor has no effect. The null hypothesis is the opposite of the alternative hypothesis. The null hypothesis is what we are testing when we perform a hypothesis test.

The output of a statistical test like the t-test is a p -value. A p -value is the probability that any effects we see in the sampled data are the result of random sampling error (chance).

If a p -value is greater than the significance level (0.05 for 5% significance) we fail to reject the null hypothesis since there is a significant possibility that our results falsify our alternative hypothesis.
If a p -value is lower than the significance level (0.05 for 5% significance) we reject the null hypothesis and have corroborated (provided evidence for) our alternative hypothesis.

The calculation and interpretation of the p -value goes back to the central limit theorem , which states that random sampling error has a normal distribution.

Using our example of a clinical drug trial, if the mean recovery times for the two groups are close enough together that there is a significant possibility ( p > 0.05) that the recovery times are the same (falsification), we fail to reject the null hypothesis.

However, if the mean recovery times for the two groups are far enough apart that the probability they are the same is under the level of significance ( p < 0.05), we reject the null hypothesis and have corroborated our alternative hypothesis.

Significance means that an effect is "probably caused by something other than mere chance" (Merriam-Webster 2022) .

The significance level (α) is the threshold for significance and, by convention, is usually 5%, 10%, or 1%, which corresponds to 95% confidence, 90% confidence, or 99% confidence, respectively.
A factor is considered statistically significant if the probability that the effect we see in the data is a result of random sampling error (the p -value) is below the chosen significance level.
A statistical test is used to evaluate whether a factor being considered is statistically significant (Gallo 2016) .

Type I vs. Type II Errors

Although we are making a binary choice between rejecting and failing to reject the null hypothesis, because we are using sampled data, there is always the possibility that the choice we have made is an error.

There are two types of errors that can occur in hypothesis testing.

Type I error (false positive) occurs when a low p -value causes us to reject the null hypothesis, but the factor does not actually result in the effect.
Type II error (false negative) occurs when a high p -value causes us to fail to reject the null hypothesis, but the factor does actually result in the effect.

The numbering of the errors reflects the predisposition of the scientific method to be fundamentally skeptical . Accepting a fact about the world as true when it is not true is considered worse than rejecting a fact about the world that actually is true.

Statistical Significance vs. Importance

When we fail to reject the null hypothesis, we have found information that is commonly called statistically significant . But there are multiple challenges with this terminology.

First, statistical significance is distinct from importance (NIST 2012) . For example, if sampled data reveals a statistically significant difference in cancer rates, that does not mean that the increased risk is important enough to justify expensive mitigation measures. All statistical results require critical interpretation within the context of the phenomenon being observed. People with different values and incentives can have different interpretations of whether statistically significant results are important.

Second, the use of 95% probability for defining confidence intervals is an arbitrary convention. This creates a good vs. bad binary that suggests a "finality and certitude that are rarely justified." Alternative approaches like Beyesian statistics that express results as probabilities can offer more nuanced ways of dealing with complexity and uncertainty (Clayton 2022) .

Science vs. Non-science

Not all ideas can be falsified, and Popper uses the distinction between falsifiable and non-falsifiable ideas to make a distinction between science and non-science. In order for an idea to be science it must be an idea that can be demonstrated to be false.

While Popper asserts there is still value in ideas that are not falsifiable, such ideas are not science in his conception of what science is. Such non-science ideas often involve questions of subjective values or unseen forces that are complex, amorphous, or difficult to objectively observe.

Falsifiable (Science)	Non-Falsifiable (Non-Science)
Murder death rates by firearms tend to be higher in countries with higher gun ownership rates	Murder is wrong
Marijuana users may be more likely than nonusers to	The benefits of marijuana outweigh the risks
Job candidates who meaningfully research the companies they are interviewing with have higher success rates	Prayer improves success in job interviews

Example Data

As example data, this tutorial will use a table of anonymized individual responses from the CDC's Behavioral Risk Factor Surveillance System . The BRFSS is a "system of health-related telephone surveys that collect state data about U.S. residents regarding their health-related risk behaviors, chronic health conditions, and use of preventive services" (CDC 2019) .

A CSV file with the selected variables used in this tutorial is available here and can be imported into R with read.csv() .

Guidance on how to download and process this data directly from the CDC website is available here...

Variable Types

The publicly-available BRFSS data contains a wide variety of discrete, ordinal, and categorical variables. Variables often contain special codes for non-responsiveness or missing (NA) values. Examples of how to clean these variables are given here...

The BRFSS has a codebook that gives the survey questions associated with each variable, and the way that responses are encoded in the variable values.

Normality Tests

Tests are commonly divided into two groups depending on whether they are built on the assumption that the continuous variable has a normal distribution.

Parametric tests presume a normal distribution.
Non-parametric tests can work with normal and non-normal distributions.

The distinction between parametric and non-parametric techniques is especially important when working with small numbers of samples (less than 40 or so) from a larger population.

The normality tests given below do not work with large numbers of values, but with many statistical techniques, violations of normality assumptions do not cause major problems when large sample sizes are used. (Ghasemi and Sahediasi 2012) .

The Shapiro-Wilk Normality Test

Data: A continuous or discrete sampled variable
R Function: shapiro.test()
Null hypothesis (H 0 ): The population distribution from which the sample is drawn is not normal
History: Samuel Sanford Shapiro and Martin Wilk (1965)

This is an example with random values from a normal distribution.

This is an example with random values from a uniform (non-normal) distribution.

The Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov is a more-generalized test than the Shapiro-Wilks test that can be used to test whether a sample is drawn from any type of distribution.

Data: A continuous or discrete sampled variable and a reference probability distribution
R Function: ks.test()
Null hypothesis (H 0 ): The population distribution from which the sample is drawn does not match the reference distribution
History: Andrey Kolmogorov (1933) and Nikolai Smirnov (1948)
pearson.test() The Pearson Chi-square Normality Test from the nortest library. Lower p-values (closer to 0) means to reject the reject the null hypothesis that the distribution IS normal.

Modality Tests of Samples

Comparing two central tendencies: tests with continuous / discrete data, one sample t-test (two-sided).

The one-sample t-test tests the significance of the difference between the mean of a sample and an expected mean.

Data: A continuous or discrete sampled variable and a single expected mean (μ)
Parametric (normal distributions)
R Function: t.test()
Null hypothesis (H 0 ): The means of the sampled distribution matches the expected mean.
History: William Sealy Gosset (1908)

t = ( Χ - μ) / (σ̂ / √ n )

t : The value of t used to find the p-value
Χ : The sample mean
μ: The population mean
σ̂: The estimate of the standard deviation of the population (usually the stdev of the sample
n : The sample size

T-tests should only be used when the population is at least 20 times larger than its respective sample. If the sample size is too large, the low p-value makes the insignificant look significant. .

For example, we test a hypothesis that the mean weight in IL in 2020 is different than the 2005 continental mean weight.

Walpole et al. (2012) estimated that the average adult weight in North America in 2005 was 178 pounds. We could presume that Illinois is a comparatively normal North American state that would follow the trend of both increased age and increased weight (CDC 2021) .

The low p-value leads us to reject the null hypothesis and corroborate our alternative hypothesis that mean weight changed between 2005 and 2020 in Illinois.

One Sample T-Test (One-Sided)

Because we were expecting an increase, we can modify our hypothesis that the mean weight in 2020 is higher than the continental weight in 2005. We can perform a one-sided t-test using the alternative="greater" parameter.

The low p-value leads us to again reject the null hypothesis and corroborate our alternative hypothesis that mean weight in 2020 is higher than the continental weight in 2005.

Note that this does not clearly evaluate whether weight increased specifically in Illinois, or, if it did, whether that was caused by an aging population or decreasingly healthy diets. Hypotheses based on such questions would require more detailed analysis of individual data.

Although we can see that the mean cancer incidence rate is higher for counties near nuclear plants, there is the possiblity that the difference in means happened by accident and the nuclear plants have nothing to do with those higher rates.

The t-test allows us to test a hypothesis. Note that a t-test does not "prove" or "disprove" anything. It only gives the probability that the differences we see between two areas happened by chance. It also does not evaluate whether there are other problems with the data, such as a third variable, or inaccurate cancer incidence rate estimates.

Note that this does not prove that nuclear power plants present a higher cancer risk to their neighbors. It simply says that the slightly higher risk is probably not due to chance alone. But there are a wide variety of other other related or unrelated social, environmental, or economic factors that could contribute to this difference.

Box-and-Whisker Chart

One visualization commonly used when comparing distributions (collections of numbers) is a box-and-whisker chart. The boxes show the range of values in the middle 25% to 50% to 75% of the distribution and the whiskers show the extreme high and low values.

Although Google Sheets does not provide the capability to create box-and-whisker charts, Google Sheets does have candlestick charts , which are similar to box-and-whisker charts, and which are normally used to display the range of stock price changes over a period of time.

This video shows how to create a candlestick chart comparing the distributions of cancer incidence rates. The QUARTILE() function gets the values that divide the distribution into four equally-sized parts. This shows that while the range of incidence rates in the non-nuclear counties are wider, the bulk of the rates are below the rates in nuclear counties, giving a visual demonstration of the numeric output of our t-test.

While categorical data can often be reduced to dichotomous data and used with proportions tests or t-tests, there are situations where you are sampling data that falls into more than two categories and you would like to make hypothesis tests about those categories. This tutorial describes a group of tests that can be used with that type of data.

Two-Sample T-Test

When comparing means of values from two different groups in your sample, a two-sample t-test is in order.

The two-sample t-test tests the significance of the difference between the means of two different samples.

Two normally-distributed, continuous or discrete sampled variables, OR
A normally-distributed continuous or sampled variable and a parallel dichotomous variable indicating what group each of the values in the first variable belong to
Null hypothesis (H 0 ): The means of the two sampled distributions are equal.

For example, given the low incomes and delicious foods prevalent in Mississippi, we might presume that average weight in Mississippi would be higher than in Illinois.

We test a hypothesis that the mean weight in IL in 2020 is less than the 2020 mean weight in Mississippi.

The low p-value leads us to reject the null hypothesis and corroborate our alternative hypothesis that mean weight in Illinois is less than in Mississippi.

While the difference in means is statistically significant, it is small (182 vs. 187), which should lead to caution in interpretation that you avoid using your analysis simply to reinforce unhelpful stigmatization.

Wilcoxen Rank Sum Test (Mann-Whitney U-Test)

The Wilcoxen rank sum test tests the significance of the difference between the means of two different samples. This is a non-parametric alternative to the t-test.

Data: Two continuous sampled variables
Non-parametric (normal or non-normal distributions)
R Function: wilcox.test()
Null hypothesis (H 0 ): For randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X.
History: Frank Wilcoxon (1945) and Henry Mann and Donald Whitney (1947)

The test is is implemented with the wilcox.test() function.

When the test is performed on one sample in comparison to an expected value around which the distribution is symmetrical (μ), the test is known as a Mann-Whitney U test .
When the test is performed to compare two samples, the test is known as a Wilcoxon rank sum test .

For this example, we will use AVEDRNK3: During the past 30 days, on the days when you drank, about how many drinks did you drink on the average?

1 - 76: Number of drinks
77: Don’t know/Not sure
99: Refused
NA: Not asked or Missing

The histogram clearly shows this to be a non-normal distribution.

Continuing the comparison of Illinois and Mississippi from above, we might presume that with all that warm weather and excellent food in Mississippi, they might be inclined to drink more. The means of average number of drinks per month seem to suggest that Mississippians do drink more than Illinoians.

We can test use wilcox.test() to test a hypothesis that the average amount of drinking in Illinois is different than in Mississippi. Like the t-test, the alternative can be specified as two-sided or one-sided, and for this example we will test whether the sampled Illinois value is indeed less than the Mississippi value.

The low p-value leads us to reject the null hypothesis and corroborates our hypothesis that average drinking is lower in Illinois than in Mississippi. As before, this tells us nothing about why this is the case.

Weighted Two-Sample T-Test

The downloadable BRFSS data is raw, anonymized survey data that is biased by uneven geographic coverage of survey administration (noncoverage) and lack of responsiveness from some segments of the population (nonresponse). The X_LLCPWT field (landline, cellphone weighting) is a weighting factor added by the CDC that can be assigned to each response to compensate for these biases.

The wtd.t.test() function from the weights library has a weights parameter that can be used to include a weighting factor as part of the t-test.

Comparing Proportions: Tests with Categorical Data

Chi-squared goodness of fit.

Tests the significance of the difference between sampled frequencies of different values and expected frequencies of those values
Data: A categorical sampled variable and a table of expected frequencies for each of the categories
R Function: chisq.test()
Null hypothesis (H 0 ): The relative proportions of categories in one variable are different from the expected proportions
History: Karl Pearson (1900)
Example Question: Are the voting preferences of voters in my district significantly different from the current national polls?

For example, we test a hypothesis that smoking rates changed between 2000 and 2020.

In 2000, the estimated rate of adult smoking in Illinois was 22.3% (Illinois Department of Public Health 2004) .

The variable we will use is SMOKDAY2: Do you now smoke cigarettes every day, some days, or not at all?

1: Current smoker - now smokes every day
2: Current smoker - now smokes some days
3: Not at all
7: Don't know
NA: Not asked or missing - NA is used for people who have never smoked

We subset only yes/no responses in Illinois and convert into a dummy variable (yes = 1, no = 0).

The listing of the table as percentages indicates that smoking rates were halved between 2000 and 2020, but since this is sampled data, we need to run a chi-squared test to make sure the difference can't be explained by the randomness of sampling.

In this case, the very low p-value leads us to reject the null hypothesis and corroborates the alternative hypothesis that smoking rates changed between 2000 and 2020.

Chi-Squared Contingency Analysis / Test of Independence

Tests the significance of the difference between frequencies between two different groups
Data: Two categorical sampled variables
Null hypothesis (H 0 ): The relative proportions of one variable are independent of the second variable.

We can also compare categorical proportions between two sets of sampled categorical variables.

The chi-squared test can is used to determine if two categorical variables are independent. What is passed as the parameter is a contingency table created with the table() function that cross-classifies the number of rows that are in the categories specified by the two categorical variables.

The null hypothesis with this test is that the two categories are independent. The alternative hypothesis is that there is some dependency between the two categories.

For this example, we can compare the three categories of smokers (daily = 1, occasionally = 2, never = 3) across the two categories of states (Illinois and Mississippi).

The low p-value leads us to reject the null hypotheses that the categories are independent and corroborates our hypotheses that smoking behaviors in the two states are indeed different.

p-value = 1.516e-09

Weighted Chi-Squared Contingency Analysis

As with the weighted t-test above, the weights library contains the wtd.chi.sq() function for incorporating weighting into chi-squared contingency analysis.

As above, the even lower p-value leads us to again reject the null hypothesis that smoking behaviors are independent in the two states.

Suppose that the Macrander campaign would like to know how partisan this election is. If people are largely choosing to vote along party lines, the campaign will seek to get their base voters out to the polls. If people are splitting their ticket, the campaign may focus their efforts more broadly.

In the example below, the Macrander campaign took a small poll of 30 people asking who they wished to vote for AND what party they most strongly affiliate with.

The output of table() shows fairly strong relationship between party affiliation and candidates. Democrats tend to vote for Macrander, while Republicans tend to vote for Stewart, while independents all vote for Miller.

This is reflected in the very low p-value from the chi-squared test. This indicates that there is a very low probability that the two categories are independent. Therefore we reject the null hypothesis.

In contrast, suppose that the poll results had showed there were a number of people crossing party lines to vote for candidates outside their party. The simulated data below uses the runif() function to randomly choose 50 party names.

The contingency table() shows no clear relationship between party affiliation and candidate. This is validated quantitatively by the chi-squared test. The fairly high p-value of 0.4018 indicates a 40% chance that the two categories are independent. Therefore, we fail to reject the null hypothesis and the campaign should focus their efforts on the broader electorate.

The warning message given by the chisq.test() function indicates that the sample size is too small to make an accurate analysis. The simulate.p.value = T parameter adds Monte Carlo simulation to the test to improve the estimation and get rid of the warning message. However, the best way to get rid of this message is to get a larger sample.

Comparing Categorical and Continuous Variables

Analysis of variation (anova).

Analysis of Variance (ANOVA) is a test that you can use when you have a categorical variable and a continuous variable. It is a test that considers variability between means for different categories as well as the variability of observations within groups.

There are a wide variety of different extensions of ANOVA that deal with covariance (ANCOVA), multiple variables (MANOVA), and both of those together (MANCOVA). These techniques can become quite complicated and also assume that the values in the continuous variables have a normal distribution.

Data: One or more categorical (independent) variables and one continuous (dependent) sampled variable
R Function: aov()
Null hypothesis (H 0 ): There is no difference in means of the groups defined by each level of the categorical (independent) variable
History: Ronald Fisher (1921)
Example Question: Do low-, middle- and high-income people vary in the amount of time they spend watching TV?

As an example, we look at the continuous weight variable (WEIGHT2) split into groups by the eight income categories in INCOME2: Is your annual household income from all sources?

1: Less than $10,000
2: $10,000 to less than $15,000
3: $15,000 to less than $20,000
4: $20,000 to less than $25,000
5: $25,000 to less than $35,000
6: $35,000 to less than $50,000
7: $50,000 to less than $75,000)
8: $75,000 or more

The barplot() of means does show variation among groups, although there is no clear linear relationship between income and weight.

To test whether this variation could be explained by randomness in the sample, we run the ANOVA test.

The low p-value leads us to reject the null hypothesis that there is no difference in the means of the different groups, and corroborates the alternative hypothesis that mean weights differ based on income group.

However, it gives us no clear model for describing that relationship and offers no insights into why income would affect weight, especially in such a nonlinear manner.

Suppose you are performing research into obesity in your city. You take a sample of 30 people in three different neighborhoods (90 people total), collecting information on health and lifestyle. Two variables you collect are height and weight so you can calculate body mass index . Although this index can be misleading for some populations (notably very athletic people), ordinary sedentary people can be classified according to BMI:

Average BMI in the US from 2007-2010 was around 28.6 and rising, standard deviation of around 5 .

You would like to know if there is a difference in BMI between different neighborhoods so you can know whether to target specific neighborhoods or make broader city-wide efforts. Since you have more than two groups, you cannot use a t-test().

Kruskal-Wallace One-Way Analysis of Variance

A somewhat simpler test is the Kruskal-Wallace test which is a nonparametric analogue to ANOVA for testing the significance of differences between two or more groups.

R Function: kruskal.test()
Null hypothesis (H 0 ): The samples come from the same distribution.
History: William Kruskal and W. Allen Wallis (1952)

For this example, we will investigate whether mean weight varies between the three major US urban states: New York, Illinois, and California.

To test whether this variation could be explained by randomness in the sample, we run the Kruskal-Wallace test.

The low p-value leads us to reject the null hypothesis that the samples come from the same distribution. This corroborates the alternative hypothesis that mean weights differ based on state.

A convienent way of visualizing a comparison between continuous and categorical data is with a box plot , which shows the distribution of a continuous variable across different groups:

A percentile is the level at which a given percentage of the values in the distribution are below: the 5th percentile means that five percent of the numbers are below that value.

The quartiles divide the distribution into four parts. 25% of the numbers are below the first quartile. 75% are below the third quartile. 50% are below the second quartile, making it the median.

Box plots can be used with both sampled data and population data.

The first parameter to the box plot is a formula: the continuous variable as a function of (the tilde) the second variable. A data= parameter can be added if you are using variables in a data frame.

The chi-squared test can be used to determine if two categorical variables are independent of each other.

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$\mu$

Four Step Process of Hypothesis Testing

There are 4 major steps in hypothesis testing:

State the hypothesis- This step is started by stating null and alternative hypothesis which is presumed as true.
Formulate an analysis plan and set the criteria for decision- In this step, a significance level of test is set. The significance level is the probability of a false rejection in a hypothesis test.
Analyze sample data- In this, a test statistic is used to formulate the statistical comparison between the sample mean and the mean of the population or standard deviation of the sample and standard deviation of the population.
Interpret decision- The value of the test statistic is used to make the decision based on the significance level. For example, if the significance level is set to 0.1 probability, then the sample mean less than 10% will be rejected. Otherwise, the hypothesis is retained to be true.

One Sample T-Testing

One sample T-Testing approach collects a huge amount of data and tests it on random samples. To perform T-Test in R, normally distributed data is required. This test is used to test the mean of the sample with the population. For example, the height of persons living in an area is different or identical to other persons living in other areas.

Syntax: t.test(x, mu) Parameters: x: represents numeric vector of data mu: represents true value of the mean

To know about more optional parameters of t.test() , try the below command:

Example:

Data: The dataset ‘x’ was used for the test.
The determined t-value is -49.504.
Degrees of Freedom (df): The t-test has 99 degrees of freedom.
The p-value is 2.2e-16, which indicates that there is substantial evidence refuting the null hypothesis.
Alternative hypothesis: The true mean is not equal to five, according to the alternative hypothesis.
95 percent confidence interval: (-0.1910645, 0.2090349) is the confidence interval’s value. This range denotes the values that, with 95% confidence, correspond to the genuine population mean.

Two Sample T-Testing

In two sample T-Testing, the sample vectors are compared. If var. equal = TRUE, the test assumes that the variances of both the samples are equal.

Syntax: t.test(x, y) Parameters: x and y: Numeric vectors

Directional Hypothesis

Using the directional hypothesis, the direction of the hypothesis can be specified like, if the user wants to know the sample mean is lower or greater than another mean sample of the data.

Syntax: t.test(x, mu, alternative) Parameters: x: represents numeric vector data mu: represents mean against which sample data has to be tested alternative: sets the alternative hypothesis

One Sample -Test

This type of test is used when comparison has to be computed on one sample and the data is non-parametric. It is performed using wilcox.test() function in R programming.

Syntax: wilcox.test(x, y, exact = NULL) Parameters: x and y: represents numeric vector exact: represents logical value which indicates whether p-value be computed

To know about more optional parameters of wilcox.test() , use below command:

The calculated test statistic or V value is 2555.
P-value: The null hypothesis is weakly supported by the p-value of 0.9192.
The alternative hypothesis asserts that the real location is not equal to 0. This indicates that there is a reasonable suspicion that the distribution’s median or location parameter is different from 0.

Two Sample -Test

This test is performed to compare two samples of data. Example:

Correlation Test

This test is used to compare the correlation of the two vectors provided in the function call or to test for the association between the paired samples.

Syntax: cor.test(x, y) Parameters: x and y: represents numeric data vectors

To know about more optional parameters in cor.test() function, use below command:

Data: The variables’mtcars$mpg’ and’mtcars$hp’ from the ‘mtcars’ dataset were subjected to a correlation test.
t-value: The t-value that was determined is -6.7424.
Degrees of Freedom (df): The test has 30 degrees of freedom.
The p-value is 1.788e-07, indicating that there is substantial evidence that rules out the null hypothesis.
The alternative hypothesis asserts that the true correlation is not equal to 0, indicating that “mtcars$mpg” and “mtcars$hp” are significantly correlated.
95 percent confidence interval: (-0.8852686, -0.5860994) is the confidence interval. This range denotes the values that, with a 95% level of confidence, represent the genuine population correlation coefficient.
Correlation coefficient sample estimate: The correlation coefficient sample estimate is -0.7761684.

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Introduction to Statistics with R

6.2 hypothesis tests, 6.2.1 illustrating a hypothesis test.

Let’s say we have a batch of chocolate bars, and we’re not sure if they are from Theo’s. What can the weight of these bars tell us about the probability that these are Theo’s chocolate?

Now, let’s perform a hypothesis test on this chocolate of an unknown origin.

What is the sampling distribution of the bar weight under the null hypothesis that the bars from Theo’s weigh 40 grams on average? We’ll need to specify the standard deviation to obtain the sampling distribution, and here we’ll use $\sigma_X = 2$ (since that’s the value we used for the distribution we sampled from).

The null hypothesis is \[H_0: \mu = 40\] since we know the mean weight of Theo’s chocolate bars is 40 grams.

The sample distribution of the sample mean is: \[ \overline{X} \sim {\cal N}\left(\mu, \frac{\sigma}{\sqrt{n}}\right) = {\cal N}\left(40, \frac{2}{\sqrt{20}}\right). \] We can visualize the situation by plotting the p.d.f. of the sampling distribution under $H_0$ along with the location of our observed sample mean.

6.2.2 Hypothesis Tests for Means

6.2.2.1 known standard deviation.

It is simple to calculate a hypothesis test in R (in fact, we already implicitly did this in the previous section). When we know the population standard deviation, we use a hypothesis test based on the standard normal, known as a $z$ -test. Here, let’s assume $\sigma_X = 2$ (because that is the standard deviation of the distribution we simulated from above) and specify the alternative hypothesis to be \[ H_A: \mu \neq 40. \] We will the z.test() function from the BSDA package, specifying the confidence level via conf.level , which is $1 - \alpha = 1 - 0.05 = 0.95$ , for our test:

6.2.2.2 Unknown Standard Deviation

If we do not know the population standard deviation, we typically use the t.test() function included in base R. We know that: \[\frac{\overline{X} - \mu}{\frac{s_x}{\sqrt{n}}} \sim t_{n-1},\] where $t_{n-1}$ denotes Student’s $t$ distribution with $n - 1$ degrees of freedom. We only need to supply the confidence level here:

We note that the $p$ -value here (rounded to 4 decimal places) is 0.0031, so again, we can detect it’s not likely that these bars are from Theo’s. Even with a very small sample, the difference is large enough (and the standard deviation small enough) that the $t$ -test can detect it.

6.2.3 Two-sample Tests

6.2.3.1 unpooled two-sample t-test.

Now suppose we have two batches of chocolate bars, one of size 40 and one of size 45. We want to test whether they come from the same factory. However we have no information about the distributions of the chocolate bars. Therefore, we cannot conduct a one sample t-test like above as that would require some knowledge about $\mu_0$ , the population mean of chocolate bars.

We will generate the samples from normal distribution with mean 45 and 47 respectively. However, let’s assume we do not know this information. The population standard deviation of the distributions we are sampling from are both 2, but we will assume we do not know that either. Let us denote the unknown true population means by $\mu_1$ and $\mu_2$ .

Consider the test $H_0:\mu_1=\mu_2$ versus $H_1:\mu_1\neq\mu_2$ . We can use R function t.test again, since this function can perform one- and two-sided tests. In fact, t.test assumes a two-sided test by default, so we do not have to specify that here.

The p-value is much less than .05, so we can quite confidently reject the null hypothesis. Indeed, we know from simulating the data that $\mu_1\neq\mu_2$ , so our test led us to the correct conclusion!

Consider instead testing $H_0:\mu_1=\mu_2$ versus $H_1:\mu_1\leq\mu_2$ .

As we would expect, this test also rejects the null hypothesis. One-sided tests are more common in practice as they provide a more principled description of the relationship between the datasets. For example, if you are comparing your new drug’s performance to a “gold standard”, you really only care if your drug’s performance is “better” (a one-sided alternative), and not that your drug’s performance is merely “different” (a two-sided alternative).

6.2.3.2 Pooled Two-sample t-test

Suppose you knew that the samples are coming from distributions with same standard deviations. Then it makes sense to carry out a pooled 2 sample t-test. You specify this in the t.test function as follows.

6.2.3.3 Paired t-test

Suppose we take a batch of chocolate bars and stamp the Theo’s logo on them. We want to know if the stamping process significantly changes the weight of the chocolate bars. Let’s suppose that the true change in weight is distributed as a ${\cal N}(-0.3, 0.2^2)$ random variable:

Let $\mu_1$ and $\mu_2$ be the true means of the distributions of chocolate weights before and after the stamping process. Suppose we want to test $H_0:\mu_1=\mu_2$ versus $\mu_1\neq\mu_2$ . We can use the R function t.test() for this by choosing paired = TRUE , which indicates that we are looking at pairs of observations corresponding to the same experimental subject and testing whether or not the difference in distribution means is zero.

We can also perform the same test as a one sample t-test using choc.after - choc.batch .

Notice that we get the exact same $p$ -value for these two tests.

Since the p-value is less than .05, we reject the null hypothesis at level .05. Hence, we have enough evidence in the data to claim that stamping a chocolate bar significantly reduces its weight.

6.2.4 Tests for Proportions

Let’s look at the proportion of Theo’s chocolate bars with a weight exceeding 38g:

Going back to that first batch of 20 chocolate bars of unknown origin, let’s see if we can test whether they’re from Theo’s based on the proportion weighing > 38g.

Recall from our test on the means that we rejected the null hypothesis that the means from the two batches were equal. In this case, a one-sided test is appropiate, and our hypothesis is:

Null hypothesis: $H_0: p = 0.85$ . Alternative: $H_A: p > 0.85$ .

We want to test this hypothesis at a level $\alpha = 0.05$ .

In R, there is a function called prop.test() that you can use to perform tests for proportions. Note that prop.test() only gives you an approximate result.

Similarly, you can use the binom.test() function for an exact result.

The $p$ -value for both tests is around 0.18, which is much greater than 0.05. So, we cannot reject the hypothesis that the unknown bars come from Theo’s. This is not because the tests are less accurate than the ones we ran before, but because we are testing a less sensitive measure: the proportion weighing > 38 grams, rather than the mean weights. Also, note that this doesn’t mean that we can conclude that these bars do come from Theo’s – why not?

The prop.test() function is the more versatile function in that it can deal with contingency tables, larger number of groups, etc. The binom.test() function gives you exact results, but you can only apply it to one-sample questions.

6.2.5 Power

Let’s think about when we reject the null hypothesis. We would reject the null hypothesis if we observe data with too small of a $p$ -value. We can calculate the critical value where we would reject the null if we were to observe data that would lead to a more extreme value.

Suppose we take a sample of chocolate bars of size n = 20 , and our null hypothesis is that the bars come from Theo’s ( $H_0$ : mean = 40, sd = 2 ). Then for a one-sided test (versus larger alternatives), we can calculate the critical value by using the quantile function in R, specifiying the mean and sd of the sampling distribution of $\overline X$ under $H_0$ :

Now suppose we want to calculate the power of our hypothesis test: the probability of rejecting the null hypothesis when the null hypothesis is false. In order to do so, we need to compare the null to a specific alternative, so we choose $H_A$ : mean = 42, sd = 2 . Then the probability that we reject the null under this specific alternative is

We can use R to perform the same calculations using the power.z.test from the asbio package:

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Introduction to Hypothesis Testing in R – Learn every concept from Scratch!

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This tutorial is all about hypothesis testing in R. First, we will introduce you with the statistical hypothesis in R, subsequently, we will cover the decision error in R, one and two-sample t-test, μ-test, correlation and covariance in R, etc.

Introduction to Statistical Hypothesis Testing in R

A statistical hypothesis is an assumption made by the researcher about the data of the population collected for any experiment. It is not mandatory for this assumption to be true every time. Hypothesis testing, in a way, is a formal process of validating the hypothesis made by the researcher.

In order to validate a hypothesis, it will consider the entire population into account. However, this is not possible practically. Thus, to validate a hypothesis, it will use random samples from a population. On the basis of the result from testing over the sample data, it either selects or rejects the hypothesis.

Statistical Hypothesis Testing can be categorized into two types as below:

Null Hypothesis – Hypothesis testing is carried out in order to test the validity of a claim or assumption that is made about the larger population. This claim that involves attributes to the trial is known as the Null Hypothesis. The null hypothesis testing is denoted by H0 .
Alternative Hypothesis – An alternative hypothesis would be considered valid if the null hypothesis is fallacious. The evidence that is present in the trial is basically the data and the statistical computations that accompany it. The alternative hypothesis testing is denoted by H 1 or H a .

Let’s take an example of the coin. We want to conclude that a coin is unbiased or not. Since null hypothesis refers to the natural state of an event, thus, according to the null hypothesis, there would an equal number of occurrences of heads and tails, if a coin is tossed several times. On the other hand, the alternative hypothesis negates the null hypothesis and refers that the occurrences of heads and tails would have significant differences in number.

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Hypothesis Testing in R

Statisticians use hypothesis testing to formally check whether the hypothesis is accepted or rejected. Hypothesis testing is conducted in the following manner:

State the Hypotheses – Stating the null and alternative hypotheses.
Formulate an Analysis Plan – The formulation of an analysis plan is a crucial step in this stage.
Analyze Sample Data – Calculation and interpretation of the test statistic, as described in the analysis plan.
Interpret Results – Application of the decision rule described in the analysis plan.

Hypothesis testing ultimately uses a p-value to weigh the strength of the evidence or in other words what the data are about the population. The p-value ranges between 0 and 1. It can be interpreted in the following way:

A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject it.
A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject it.

A p-value very close to the cutoff (0.05) is considered to be marginal and could go either way.

Decision Errors in R

The two types of error that can occur from the hypothesis testing:

Type I Error – Type I error occurs when the researcher rejects a null hypothesis when it is true. The term significance level is used to express the probability of Type I error while testing the hypothesis. The significance level is represented by the symbol α (alpha).
Type II Error – Accepting a false null hypothesis H 0 is referred to as the Type II error. The term power of the test is used to express the probability of Type II error while testing hypothesis. The power of the test is represented by the symbol β (beta).

Using the Student’s T-test in R

The Student’s T-test is a method for comparing two samples. It can be implemented to determine whether the samples are different. This is a parametric test, and the data should be normally distributed.

R can handle the various versions of T-test using the t.test() command. The test can be used to deal with two- and one-sample tests as well as paired tests.

Listed below are the commands used in the Student’s t-test and their explanation:

t.test(data.1, data.2) – The basic method of applying a t-test is to compare two vectors of numeric data.
var.equal = FALSE – If the var.equal instruction is set to TRUE, the variance is considered to be equal and the standard test is carried out. If the instruction is set to FALSE (the default), the variance is considered unequal and the Welch two-sample test is carried out.
mu = 0 – If a one-sample test is carried out, mu indicates the mean against which the sample should be tested.
alternative = “two.sided” – It sets the alternative hypothesis. The default value for this is “two.sided” but a greater or lesser value can also be assigned. You can abbreviate the instruction.
conf.level = 0.95 – It sets the confidence level of the interval (default = 0.95).
paired = FALSE – If set to TRUE, a matched pair T-test is carried out.
t.test(y ~ x, data, subset) – The required data can be specified as a formula of the form response ~ predictor. In this case, the data should be named and a subset of the predictor variable can be specified.
subset = predictor %in% c(“sample.1”, sample.2”) – If the data is in the form response ~ predictor, the two samples to be selected from the predictor should be specified by the subset instruction from the column of the data.

Two-Sample T-test with Unequal Variance

The t.test() command is generally used to compare two vectors of numeric values. The vectors can be specified in a variety of ways, depending on how your data objects are set out.

The default form of the t.test() command does not assume that the samples have equal variance. As a result, the two-sample test is carried out unless specified otherwise. The two-sample test can be on any two datasets using the following command:

The default clause in the t.test() command can be overridden. To do so, add the var.equal = TRUE. This is an instruction that is added to the t.test() command. This instruction forces the t.test() command to assume that the variance of the two samples is equal.

The magnitude of the degree of freedom is unmodified as well as the calculations of t-value makes use of the pooled variance.

As a result, the p-value is slightly different from the Welch version. For example:

As per the samples estimate, the default clause in the t.test() command can be overridden. To do so, add the var.equal = TRUE instruction to the standard t.test() command. This instruction forces the t.test() command to assume that the variance of the two samples is equal.

One-Sample T-testing in R

To perform analysis, it collects a large amount of data from various sources and tests it on random samples. In several situations, when the population of collected data is unknown, researchers test samples to identify the population. The one-sample T-test is one of the useful tests for testing the sample’s population.

This test is used for testing the mean of samples. For example, you can use this test to compare that a sample of students from a particular college is identical or different from the sample of general students. In this situation, the hypothesis tests that the sample is from a known population with a known mean (m) or from an unknown population.

To carry out a one-sample T-test in R , the name of a single vector and the mean with which it is compared is supplied.

The mean defaults to 0.

The one-sample T-test can be implemented as follows:

Learn to perform T-tests in R and master the concept

Using Directional Hypotheses in R

You can also specify a “direction” to your hypothesis.

In many cases, you are simply testing to see if the means of two samples are different, but you may want to know if a sample mean is lower or greater than another sample mean. You can use the alternative equal to (=) instruction to switch the emphasis from a two-sided test (the default) to a one-sided test. The choices you have are between ″two.sided″, ″less″, or ″greater″, and the choice can be abbreviated, as shown in the following command:

Formula Syntax and Subsetting Samples in the T-test in R

As discussed in the previous sections, the T-test is designed to compare two samples.

So far, we have seen how to carry out the T-test on separate vectors of values; however, your data may be in a more structured form with a column for the response variable and a column for the predictor variable.

When the data is available in a more structured form with a separate column for the response variable and predictor variable, the data can be set in a more sensible and flexible manner. You need a new way to deal with the layout.

R deals with the layout by using a formula syntax.

In this section, we will use the grass dataset:

You can download the dataset from here – Grass Dataset

You can create a formula by using the tilde (~) symbol. Essentially, your response variable goes to the left of the ~ and the predictor goes to the right, as shown in the following command:

If your predictor column contains more than two items, the T-test cannot be used; however, you can still carry out a test by subsetting this predictor column and specifying the two samples you want to compare.

The subset = instruction should be used as a part of the t.test() command, as follows:

Formula Syntax in R – The following example illustrates how to do this using the same data as in the previous example:

You first specify the column from which you want to take your subset and then type %in%. This tells the command that the list that follows is in the graze column. Note that, you have to put the levels in quotes; here you compare ″mow″and ″unmow″and your result is identical to the one you obtained before.

μ-test in R

When you have two samples to compare and your data is nonparametric, you can use the μ-test. This goes by various names and may be known as the Mann—Whitney μ-test or Wilcoxon sign rank test. The wilcox.test() command can carry out the analysis.

The wilcox.test() command can conduct two-sample or one-sample tests, and you can add a variety of instructions to carry out the test.

Given below are the main options available in the wilcox.test() command with their explanation:

test(sample.1, sample.2) – It carries out a basic two-sample μ-test on the numerical vectors specified.
mu = 0 – If a one-sample test is carried out, mu indicates the value against which the sample should be tested.
alternative = “two.sided” – It sets the alternative hypothesis. “two.sided” is the default value, but a greater or lesser value can also be assigned. You can abbreviate the instruction but you still need the quotes.
int = FALSE – It sets whether confidence intervals should be reported or not.
level = 0.95 – It sets the confidence level of the interval (default = 0.95).
correct = TRUE – By default, the continuity correction is applied. This can also be set to FALSE.
paired = FALSE – If set to TRUE, a matched pair μ-test is carried out.
exact = NULL – It sets whether an exact p-value should be computed. The default is to do so for less than 50 items.
test(y ~ x, data, subset) – The required data can be specified as a formula of the form response ~ predictor. In this case, the data should be named and a subset of the predictor variable can be specified.
subset = predictor %in% c(″1″, ″sample.2″) – If the data is in the form response ~ predictor, the subset instruction can specify the two samples to select from the predictor column of the data.

Don’t forget to check the R Vector Functions

Two-Sample μ-test in R

The basic way of using wilcox.test() command is to specify the two samples you want to compare as separate vectors, as shown in the following command:

By default, the confidence intervals are not calculated and the p-value is adjusted using the “continuity correction”; a message tells you that the latter has been used. In this case, you see a warning message because you have tied values in the data. If you set exact = FALSE, this message would not be displayed because the p-value would be determined from a normal approximation method.

Any doubts in Hypothesis Testing in R, till now? Share your queries in the comment section.

One-Sample μ-test in R

When you specify a single numerical vector, then it carries out a one-sample μ-test. The default is to set mu = 0. For example:

In this case, the p-value is a normal approximation because it uses the exact = FALSE instruction. The command has assumed mu = 0 because it is not specified explicitly.

Formula Syntax and Subsetting Samples in the μ-test in R

It is better to have data arranged into a data frame where one column represents the response variable and another represents the predictor variable. In this case, the formula syntax can be used to describe the situation and carry out the wilcox.test() command on your data. The method is similar to what is used for the T-test.

The basic form of the command is:

You can also use additional instructions as you could with the other syntax. If the predictor variable contains more than two samples, you cannot conduct a μ-test and use a subset that contains exactly two samples.

Notice that in the preceding command, the names of the samples must be specified in quotes in order to group them together. The μ-test is one of the most widely used statistical methods, so it is important to be comfortable in using the wilcox.test()command. In the following activity, you try conducting a range of μ-tests for yourself. The μ-test is a useful tool for comparing two samples and is one of the most widely used tools of all simple statistical tests. Both the t.test()and wilcox.test()commands can also deal with matched-pair data.

Correlation and Covariance in R

When you have two continuous variables, you can look for a link between them. This link is called a correlation.

The cor() command determines correlations between two vectors, all the columns of a data frame, or two data frames. The cov() command examines covariance. The cor.test() command carries out a test of significance of the correlation.

You can add a variety of additional instructions to these commands, as given below:

cor(x, y = NULL) – It carries out a basic correlation between x and y. If x is a matrix or data frame, we can omit y. One can correlate any object against any other object as long as the length of the individual vectors matches up.
cov(x, y = NULL) – It determines covariance between x and y. If x is a matrix or data frame, one can omit y.
cov2cor(V) – It takes a covariance matrix V and calculates the correlation.
method = – The default is “pearson”, but “spearman” or “kendall” can be specified as the methods for correlation or covariance. These can be abbreviated but you still need the quotes and note that they are lowercase.
var(x, y = NULL) – It determines the variance of x. If x is a matrix or data frame and y is specified, it also determines the covariance.
test(x, y) – It carries out a significance test of the correlation between x and y. In this case, you can now specify only two data vectors, but you can use a formula syntax, which makes it easier when the variables are within a data frame or matrix. The Pearson product-moment is the default, but it can also use Spearman’s Rho or Kendall’s Tau tests. You can use the subset command to select data on the basis of a grouping variable.
alternative = “two.sided” – The default is for a two-sided test but the alternative hypothesis can be given as “two.sided”, “greater”, or “less”.
level = 0.95 – If the method = “pearson” and n > 3, it will show the confidence intervals. This instruction sets the confidence level and defaults to 0.95.

A concept to ease your journey of R programming – R Data Frame

Simple Correlation in R

Simple correlations are between two continuous variables and use the cor() command to obtain a correlation coefficient, as shown in the following command:

This example used the Spearman Rho correlation but you can also apply Kendall’s tau by specifying method = ″kendall″. Note that you can abbreviate this but you still need the quotes. You also have to use lowercase.

If your vectors are within a data frame or some other object, you need to extract them in a different fashion.

Covariance in R

The cov() command uses syntax similar to the cor() command to examine covariance.

We can use the cov() command as:

The cov2cor() command determines the correlation from a matrix of covariance, as shown in the following command:

Significance Testing in Correlation Tests

You can apply a significance test to your correlations by using the cor.test() command. In this case, you can compare only two vectors at a time, as shown in the following command:

In the previous example, you can see that the Pearson correlation is between height and weight in the data of women and the result also shows the statistical significance of the correlation.

You must definitely learn about Descriptive Statistics in R

Formula Syntax in R

If your data is in a data frame, using the attach() or with() command is tedious, as is using the $ syntax. A formula syntax is available as an alternative, which provides a neater representation of your data, as shown in the following command:

Here you examine the data of cars, which comes built-in in R. The formula is slightly different from the one that you used previously. Here you specify both variables to the right of the ~. You also give the name of the data as a separate instruction. All the additional instructions are available while using the formula syntax as well as the subset instruction.

Tests for Association in R

When you have categorical data, you can look for associations between categories by using the chi-squared test. Routines to achieve this is possible by using the chisq.test() command.

The various additional instructions that you can add to the chisq.test() command are:

test(x, y = NULL) – A basic chi-squared test is carried out on a matrix or data frame. If it provides x as a vector, a second vector can be supplied. If x is a single vector and y is not given, a goodness of fit test is carried out.
correct = TRUE – It applies Yates’ correction if the data forms a 2 n 2 contingency table.
p = – It is a vector of probabilities for use with a goodness of fit test. If p is not given, the goodness of fit tests that the probabilities are all equal.
p = FALSE – If TRUE, p is rescaled to sum to 1. For use with the goodness of fit tests.
p.value = FALSE – If set to TRUE, a Monte Carlo simulation calculates p-values.
B = 2000 – The number of replicates to use in the Monte Carlo simulation.

Get a deep insight into Contingency Tables in R

Goodness of Fit Tests in R

While fitting a statistical model for observed data, an analyst must identify how accurately the model analysis the data. This is done with the help of the chi-square test.

The chi-square test is a type of hypothesis testing methodology that identifies the goodness-of-fit by testing whether the observed data is taken from the claimed distribution or not. The two values included in this test are observed value, the frequency of a category from the sample data, and expected frequency that is calculated on the basis of an expected distribution of the sample population. The chisq.test() command can be used to carry out the goodness of fit test.

In this case, you must have two vectors of numerical values, one representing the observed values and the other representing the expected ratio of values. The goodness of fit tests the data against the ratios you specified. If you do not specify any, the data is tested against equal probability.

The basic form of the chisq.test() command will operate on a matrix or data frame.

By enclosing the command completely within parentheses, you can get the result object to display immediately. The results of many commands are stored as a list containing several elements, and you can see what is available by using the names() command and view them by using the $syntax .

The p-value can be determined using a Monte Carlo simulation by using the simulate.p.value and B instructions. If the data form a 2 n 2 contingency, then Yates’ correction is automatically applied but only if the Monte Carlo simulation is not used.

To conduct goodness of fit test, you must specify p, the vector of probabilities; if this does not add to 1, you will get an error unless you use rescale.p = TRUE . You can use a Monte Carlo simulation on the goodness of fit test. If a single vector is specified, a goodness of fit test is carried out but the probabilities are assumed to be equal.

In this article, we studied about Hypothesis testing in R. We learned about the basics of the null hypothesis as well as alternative hypothesis. We read about T-test and μ-test. Then, we implemented these statistical methods in R.

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19 cheat sheets.

These commonly used reference sheets can also be found online. I added the links that are current as of this writing.

When you google for cheat sheets or reference sheets, make sure you check that you have the latest version. R is an evolving language. It is best to check the posit website (posit is the company that now “owns” RStudio). Many of the help sheets can be found in window (1), Help , ’Cheat sheets`. This also directs you to the posit website.

19.1 Base R reference sheets https://github.com/rstudio/cheatsheets/blob/main/base-r.pdf

19.2 RMarkdown reference sheets https://posit.co/resources/cheatsheets/?type=posit-cheatsheets/

19.3 ggplot2 reference sheets https://posit.co/resources/cheatsheets/?type=posit-cheatsheets/

19.4 dplyr cheat sheets https://posit.co/resources/cheatsheets/?type=posit-cheatsheets/

19.5 lubridate cheat sheets https://rstudio.github.io/cheatsheets/lubridate.pdf

19.6 LaTeX reference sheet https://wch.github.io/latexsheet/latexsheet.pdf

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Hypothesis testing in r.

Posted on December 3, 2022 by Jim in R bloggers | 0 Comments

The post Hypothesis Testing in R appeared first on Data Science Tutorials

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Hypothesis Testing in R, A formal statistical test called a hypothesis test is used to confirm or disprove a statistical hypothesis.

The following R hypothesis tests are demonstrated in this course.

T-test with one sample
T-Test of two samples
T-test for paired samples

Each type of test can be run using the R function t.test().

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one sample t-test

x, y: The two samples of data.

alternative: The alternative hypothesis of the test.

mu: The true value of the mean.

paired: whether or not to run a paired t-test.

var.equal: Whether to assume that the variances between the samples are equal.

conf.level: The confidence level to use.

The following examples show how to use this function in practice.

Example 1: One-Sample t-test in R

A one-sample t-test is used to determine whether the population’s mean is equal to a given value.

Consider the situation where we wish to determine whether the mean weight of a particular species of turtle is 310 pounds or not. We go out and gather a straightforward random sample of turtles with the weights listed below.

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Weights: 301, 305, 312, 315, 318, 319, 310, 318, 305, 313, 305, 305, 305

The following code shows how to perform this one sample t-test in R:

specify a turtle weights vector

Now we can perform a one-sample t-test

From the output we can see:

t-test statistic: 045145

degrees of freedom: 12

p-value: 0. 9647

95% confidence interval for true mean: [306.3644, 313.7895]

mean of turtle weights: 310.0769We are unable to reject the null hypothesis since the test’s p-value of 0. 9647 is greater than or equal to.05.

This means that we lack adequate evidence to conclude that this species of turtle’s mean weight is different from 310 pounds.

Example 2: Two Sample t-test in R

To determine whether the means of two populations are equal, a two-sample t-test is employed.

Consider the situation where we want to determine whether the mean weight of two different species of turtles is equal. We gather a straightforward random sample of turtles from each species with the following weights to test this.

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Sample 1: 310, 311, 310, 315, 311, 319, 310, 318, 315, 313, 315, 311, 313

Sample 2: 335, 339, 332, 331, 334, 339, 334, 318, 315, 331, 317, 330, 325

The following code shows how to perform this two-sample t-test in R:

Now we can create a vector of turtle weights for each sample

Let’s perform two sample t-tests

We reject the null hypothesis because the test’s p-value (6.029e-06) is smaller than.05.

Accordingly, we have enough data to conclude that the mean weight of the two species is not identical.

Example 3: Paired Samples t-test in R

When each observation in one sample can be paired with an observation in the other sample, a paired samples t-test is used to compare the means of the two samples.

For instance, let’s say we want to determine if a particular training program may help basketball players raise their maximum vertical jump (in inches).

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We may gather a small, random sample of 12 college basketball players to test this by measuring each player’s maximum vertical jump. Then, after each athlete has used the training regimen for a month, we might take another look at their max vertical leap.

The following information illustrates the maximum jump height (in inches) for each athlete before and after using the training program.

Before: 122, 124, 120, 119, 119, 120, 122, 125, 124, 123, 122, 121

After: 123, 125, 120, 124, 118, 122, 123, 128, 124, 125, 124, 120

The following code shows how to perform this paired samples t-test in R:

Let’s define before and after max jump heights

We can perform paired samples t-test

We reject the null hypothesis since the test’s p-value (0. 02803) is smaller than.05.

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The mean jump height before and after implementing the training program is not equal, thus we have enough data to conclude so.

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9 hypothesis testing cheatsheet.

Weisheng Chen

This is an PDF version of cheat sheet for hypothesis testing including key concepts, steps for conducting hypothesis testing and comparison between different tests.

Check the cheat sheet by clicking the following Github link:

https://github.com/SteveChen2751/GR5702-EDAV/blob/main/Hypothesis_Testing_Cheatsheet.pdf

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Statistical hypothesis tests return a p-value, which indicates the probability that the null hypothesis of a test is true. If the p-value is less than or equal to the significance level, then the null hypothesis is rejected in favor of the alternative hypothesis.And, if the p-value is greater than the significance level, then the null hypothesis is not rejected.
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Hypothesis Testing Cheat Sheet 23 June 2022 Hypothesis Terms Definitions Significance Level (𝜶) Defines the strength of evidence in probabilistic terms. Specifically, alpha represents the probability that tests will produce statistically significant results when the null hypothesis is correct. In most fields, α = ì. ì5 is used most often.
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This tutorial covers basic hypothesis testing in R. Normality tests. Shapiro-Wilk normality test. Kolmogorov-Smirnov test. Comparing central tendencies: Tests with continuous / discrete data. One-sample t-test : Normally-distributed sample vs. expected mean. Two-sample t-test: Two normally-distributed samples.
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6.2.2.1 Known Standard Deviation. It is simple to calculate a hypothesis test in R (in fact, we already implicitly did this in the previous section). When we know the population standard deviation, we use a hypothesis test based on the standard normal, known as a $z$-test.Here, let's assume $\sigma_X = 2$ (because that is the standard deviation of the distribution we simulated from above ...
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The Complete Guide: Hypothesis Testing in R

Example 1: One Sample t-test in R

Example 2: Two Sample t-test in R

Example 3: Paired Samples t-test in R

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Box-and-Whisker Chart

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Wilcoxen Rank Sum Test (Mann-Whitney U-Test)

Weighted Two-Sample T-Test

Comparing Proportions: Tests with Categorical Data

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Comparing Categorical and Continuous Variables

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Introduction to Statistics with R

6.2.2 Hypothesis Tests for Means

6.2.2.2 Unknown Standard Deviation

6.2.3 Two-sample Tests

6.2.3.2 Pooled Two-sample t-test

6.2.3.3 Paired t-test

6.2.4 Tests for Proportions

6.2.5 Power

Introduction to Hypothesis Testing in R – Learn every concept from Scratch!

Introduction to Statistical Hypothesis Testing in R

Hypothesis Testing in R

Decision Errors in R

Using the Student’s T-test in R

Two-Sample T-test with Unequal Variance

One-Sample T-testing in R

Using Directional Hypotheses in R

Formula Syntax and Subsetting Samples in the T-test in R

μ-test in R

Two-Sample μ-test in R