Performing T-Tests in R — Stats with R (2024)

We first want to know if it is reasonable to believe that the datawe have is normally distributed. Many people would use a histogram orstatistical test for this, while both can be valid, a preferred methodis using what is called a QQ-plot. A QQ-plot compares the density ofyour variable to the density of an ideal standard normal distribution todetermine if they are similar. I am introducing a function thatstandardizes the distribution (or turns it into z-scores) to match themean and standard deviation of the comparison distribution. What weactually care about is that the points on the graphs fall near the lineuniformly, more tips for interpreting QQ-plots can be found starting onpg.46 of my book here https://www.statswithr.com/neural-networks-with-keras-in-r-a-quickstart-guide.The plots below are representative of what you can expect a normallydistributed variable to look like with a small sample size. With asample size less than 30, points generally falling close to the line issufficient. If the data is determined to not be normal, you will need toeither transform the data or use a non-parametric version of the t-testwhich comes at a loss of statistical power (ability to find statisticalsignificance).

Lets start with the one sample t-test, where we test the hypothesisthat the population mean of Eggs is 4. We know that the population meanis actually 5 (because we set it that way), so we expect to reject thenull hypothesis assuming our sample size is sufficiently large. Below weare able to reject the null hypothesis and determine that the populationmean is likely not 4. Notice that the degrees of freedom is n-1, andthat the confidence interval does not include 4.

Before we continue with two sample t-tests, we will want to get anidea if the variances are equal. With this generated data, we know thatthey are, but we should not assume that. var.test() can give us an ideaif that assumption is met. If this test is not statisticallysignificant, the equality of variance can be assumed. If the variancesare found to likely not be equal, we can still perform t-tests, we willsimply need to assume unequal variance in the t.test() function.

Now, let’s perform a two sample t-test; the default setting assumesunequal variance, which means the degrees of freedom are reducedslightly (17.872 instead of 18). Use this if the prior test isstatistically significant with the p-value < 0.05. Note that doingthis makes it slightly more difficult to find an effect. This test wasnot statistically significant with a p-value of 0.06799, which is ofcourse above 0.05. We fail to reject the null hypothesis that thepopulation means between Eggs and Milk are equivalent.

Now, when we assume equal variance our degrees of freedom isexactly 18 or n-2. We once again fail to reject the null hypothesis;however, you will notice that the p-value here is slightly lower thanthe previous p-value by 0.00012. This means that we were just a littlecloser to rejecting the null hypothesis, which is related to the testbeing slightly more powerful in the statistical sense.

Until now we have been doing what are called two-tailed testsmeaning that we are simply testing if the population means of Milk andEggs are different in some way. If we want to test the hypothesis thatthe population mean of Eggs is greater than the population mean of Milkwe have to change the “alternative” argument of the function to greater.Note that the order of Eggs and Milk matters in this situation; if Milkwere first in the function below, we would be testing the oppositehypothesis. The t-test is finally significant with the p-value <0.05. Directional hypothesis tests have more power to find effects, butit is important that we have a prior hypothesis to justify the use of adirectional hypothesis test. Depending on the application, it is morecommon to use two-tailed hypothesis tests, as one-tailed tests can belooked upon negatively.

Finally, we will perform a paired samples t-test. The two variableswe have been using are not appropriate for this type of analysis, so Iam creating a variable for the milk price next week that depends on thecurrent milk price. Paired samples t-tests are meant to be performed ondata that are not independent samples. A common form of this is a testretest situation where the same individuals are tested twice, before andafter some manipulation. A regular t-test is inappropriate because thesamples are not independent, so we use a paired sample t-test thataccounts for the same individuals being tested twice. In this example,the milk price is being observed and then we are waiting a week andre-observing them. The p-value < 0.05, so we can say that there is astatistically significant difference between weeks, and the 95%confidence interval for the increase ranges from 0.39 and 1.76 for thepopulation means.