10.6: One-Sided Tests

When we introduced the theory of null hypothesis tests, we mentioned that there are some situations when it’s appropriate to specify a one-sided test (see Section ????.4.3). So far, all of the t-tests have been two-sided tests (as is default for SPSS and many other statistics packages). For instance, when we specified a one-sample t-test for the grades in Dr. Zeppo’s class, the null hypothesis was that the true mean was 67.5%. The alternative hypothesis was that the true mean was greater than or less than 67.5%. Suppose we were only interested in finding out if the true mean is greater than 67.5%, and have no interest whatsoever in testing to find out if the true mean is lower than 67.5%. If so, our null hypothesis would be that the true mean is 67.5% or less, and the alternative hypothesis would be that the true mean is greater than 67.5%. Newer versions of SPSS solve this issue by simply reporting both the one-sided and two-sided p-values:

Notice that although the t-statistic and degrees of freedom are not different, the p-value is. This is because the one-sided test has a different rejection region from the two-sided test. If you’ve forgotten why this is and what it means, you may find it helpful to read back over Chapter ??, and Section ??.4.3 in particular.

So that’s how to do a one-sided one sample t-test. However, all versions of the t-test can be one-sided. For an independent samples t-test, you could have a one-sided test if you’re only interested in testing to see if group A has higher scores than group B, but have no interest in finding out if group B has higher scores than group A. Let’s suppose that, for Dr. Harpo’s class, you wanted to see if Anastasia’s students had higher grades than Bernadette’s. The independentSamplesTTest() function lets you do this, again by specifying the one.sided argument. However, this time around you need to specify the name of the group that you’re expecting to have the higher score. In our case, we’d write one.sided = "Anastasia". So the command would be:

independentSamplesTTest( 
    formula = grade ~ tutor, 
    data = harpo, 
    one.sided = "Anastasia"
  )

## 
##    Welch's independent samples t-test 
## 
## Outcome variable:   grade 
## Grouping variable:  tutor 
## 
## Descriptive statistics: 
##             Anastasia Bernadette
##    mean        74.533     69.056
##    std dev.     8.999      5.775
## 
## Hypotheses: 
##    null:        population means are equal, or smaller for group 'Anastasia' 
##    alternative: population mean is larger for group 'Anastasia' 
## 
## Test results: 
##    t-statistic:  2.034 
##    degrees of freedom:  23.025 
##    p-value:  0.027 
## 
## Other information: 
##    one-sided 95% confidence interval:  [0.863, Inf] 
##    estimated effect size (Cohen's d):  0.724

Again, the output changes in a predictable way. The definition of the null and alternative hypotheses has changed, the p-value has changed, and it now reports a one-sided confidence interval rather than a two-sided one.

What about the paired samples t-test? Suppose we wanted to test the hypothesis that grades go up from test 1 to test 2 in Dr Zeppo’s class, and are not prepared to consider the idea that the grades go down. Again, we can use the one.sided argument to specify the one-sided test, and it works the same way it does for the independent samples t-test. You need to specify the name of the group whose scores are expected to be larger under the alternative hypothesis. If your data are in wide form, as they are in the chico data frame, you’d use this command:

pairedSamplesTTest( 
     formula = ~ grade_test2 + grade_test1, 
     data = chico, 
     one.sided = "grade_test2" 
  )

## 
##    Paired samples t-test 
## 
## Variables:  grade_test2 , grade_test1 
## 
## Descriptive statistics: 
##             grade_test2 grade_test1 difference
##    mean          58.385      56.980      1.405
##    std dev.       6.406       6.616      0.970
## 
## Hypotheses: 
##    null:        population means are equal, or smaller for measurement 'grade_test2' 
##    alternative: population mean is larger for measurement 'grade_test2' 
## 
## Test results: 
##    t-statistic:  6.475 
##    degrees of freedom:  19 
##    p-value:  <.001 
## 
## Other information: 
##    one-sided 95% confidence interval:  [1.03, Inf] 
##    estimated effect size (Cohen's d):  1.448

Yet again, the output changes in a predictable way. The hypotheses have changed, the p-value has changed, and the confidence interval is now one-sided. If your data are in long form, as they are in the chico2 data frame, it still works the same way. Either of the following commands would work,

> pairedSamplesTTest( 
    formula = grade ~ time, 
    data = chico2, 
    id = "id", 
    one.sided = "test2" 
  )

> pairedSamplesTTest( 
    formula = grade ~ time + (id), 
    data = chico2, 
    one.sided = "test2" 
  )

and would produce the same answer as the output shown above.