Compare the means of two or more variables or groups in the data
The compare means t-test is used to compare the mean of a variable in one group to the mean of the same variable in one, or more, other groups. The null hypothesis for the difference between the groups in the population is set to zero. We test this hypothesis using sample data.
We can perform either a one-tailed test (i.e.,
greater than) or a two-tailed test (see the ‘Alternative
hypothesis’ dropdown). We use one-tailed tests to evaluate if the
available data provide evidence that the difference in sample means
between groups is less than (or greater than ) zero.
We have access to the nine-month academic salary for Assistant Professors, Associate Professors and Professors in a college in the U.S (2008-09). The data were collected as part of an on-going effort by the college’s administration to monitor salary differences between male and female faculty members. The data has 397 observations and the following 6 variables.
The data are part of the CAR package and are linked to the book: Fox J. and Weisberg, S. (2011) An R Companion to Applied Regression, Second Edition Sage.
Suppose we want to test if professors of lower rank earn lower
salaries compared to those of higher rank. To test this hypothesis we
first select professor
rank and select
as the numerical variable to compare across ranks. In the
Choose combinations box select all available entries to
conduct pair-wise comparisons across the three levels. Note that
removing all entries will automatically select all combinations. We are
interested in a one-sided hypothesis (i.e.,
The first two blocks of output show basic information about the test (e.g., selected variables and confidence levels) and summary statistics (e.g., mean, standard deviation, margin or error, etc. per group). The final block of output shows the following:
Null hyp.is the null hypothesis and
Alt. hyp.the alternative hypothesis
diffis the difference between the sample means for two groups (e.g., 80775.99 - 93876.44 = -13100.45). If the null hypothesis is true we expect this difference to be small (i.e., close to zero)
p.valueis the probability of finding a value as extreme or more extreme than
diffif the null hypothesis is true
If we check
Show additional statistics the following
output is added:
Pairwise mean comparisons (t-test) Data : salary Variables : rank, salary Samples : independent Confidence: 0.95 Adjustment: None rank mean n n_missing sd se me AsstProf 80,775.985 67 0 8,174.113 998.627 1,993.823 AssocProf 93,876.438 64 0 13,831.700 1,728.962 3,455.056 Prof 126,772.109 266 0 27,718.675 1,699.541 3,346.322 Null hyp. Alt. hyp. diff p.value se t.value df 0% 95% AsstProf = AssocProf AsstProf < AssocProf -13100.45 < .001 1996.639 -6.561 101.286 -Inf -9785.958 *** AsstProf = Prof AsstProf < Prof -45996.12 < .001 1971.217 -23.334 324.340 -Inf -42744.474 *** AssocProf = Prof AssocProf < Prof -32895.67 < .001 2424.407 -13.569 199.325 -Inf -28889.256 *** Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
seis the standard error (i.e., the standard deviation of the sampling distribution of
t.valueis the t statistic associated with
diffthat we can compare to a t-distribution (i.e.,
dfis the degrees of freedom associated with the statistical test. Note that the Welch approximation is used for the degrees of freedom
0% 95%show the 95% confidence interval around the difference in sample means. These numbers provide a range within which the true population difference is likely to fall
There are three approaches we can use to evaluate the null hypothesis. We will choose a significance level of 0.05.1 Of course, each approach will lead to the same conclusion.
Because each of the p.values is smaller than the significance level we reject the null hypothesis for each evaluated pair of professor ranks. The data suggest that associate professors make more than assistant professors and professors make more than assistant and associate professors. Note also the ’***’ that are used as an indicator for significance.
Because zero is not contained in any of the
confidence intervals we reject the null hypothesis for each evaluated
combination of ranks. Because our alternative hypothesis is
Less than the confidence interval is actually an upper
bound for the difference in salaries in the population at a 95%
confidence level (i.e., -9785.958, -42744.474, and -28889.256)
Because the calculated t.values (-6.561, -23.334, and -13.569) are
smaller than the corresponding critical
t.value we reject the null hypothesis for each evaluated combination of
ranks. We can obtain the critical t.value by using the probability
calculator in the Basics menu. Using the test for assistant
versus associate professors as an example, we find that for a
t-distribution with 101.286 degrees of freedom (see
critical t.value is 1.66. We choose 0.05 as the lower probability bound
because the alternative hypothesis is
In addition to the numerical output provided in the Summary
tab we can also investigate the association between
salary visually (see the Plot tab). The screen
shot below shows a scatter plot of professor salaries and a bar chart
with confidence interval (black) and standard error (blue) bars.
Consistent with the results shown in the Summary tab there is
clear separation between the salaries across ranks. We could also choose
to plot the sample data as a box plot or as a set of density curves.
The more comparisons we evaluate the more likely we are to find a “significant” result just by chance even if the null hypothesis is true. If we conduct 100 tests and set our significance level at 0.05 (or 5%) we can expect to find 5 p.values smaller than or equal to 0.05 even if the are no associations in the population.
Bonferroni adjustment ensures the p.values are scaled appropriately given the number of tests conducted. This XKCD cartoon expresses the need for this type of adjustments very clearly.
This is a comparison of means test of the null hypothesis that the true population difference in means is equal to 0. Using a significance level of 0.05, we reject the null hypothesis for each pair of ranks evaluated, and conclude that the true population difference in means is less than 0.
The p.value for the test of differences in salaries between assistant and associate professors is < .001. This is the probability of observing a sample difference in means that is as or more extreme than the sample difference in means from the data if the null hypothesis is true. In this case, it is the probability of observing a sample difference in means that is less than (or equal to) -13100.45 if the true population difference in means is 0.
The 95% confidence bound is -9785.958. If repeated samples were taken and the 95% confidence bound computed for each one, the true population mean would be below the lower bound in 95% of the samples
1 The significance level, often denoted by \(\alpha\), is the highest probability you are willing to accept of rejecting the null hypothesis when it is actually true. A commonly used significance level is 0.05 (or 5%)
Add code to
> Rmd to (re)create the analysis by clicking the
icon on the bottom
left of your screen or by pressing
ALT-enter on your
If a plot was created it can be customized using
plot(result, plots = "scatter", custom = TRUE) + labs(title = "Compare means")).
> Visualize for details.
For an overview of related R-functions used by Radiant to evaluate means see Basics > Means
The key function from the
stats package used in the
compare_means tool is
Copy-and-paste the full command below into the RStudio console (i.e., the bottom-left window) and press return to gain access to all materials used in the hypothesis testing module of the Radiant Tutorial Series: