A PRIORI COMPARISONS

 

A researcher who has a theoretical basis for hypothesizing specific comparisons before a study starts may use the method of a priori comparisons.  

Twelve subjects were randomly selected and randomly assigned to one of the three conditions: the control group, the sleep deprivation group, and the food deprivation group. After receiving the treatment, all the subjects took a math test. The number of errors made in the test was recorded.  
 

Research Questions

The researcher is interested in comparing each experimental group mean with the control group mean.

1. Is there a significant difference on the dependent variable (number of errors) between the sleep deprivation group and the control group?

2. Is there a significant difference on the dependent variable (number of errors) between the food deprivation group and the control group?

 

Data Set and Descriptive Statistics

The number of errors made in solving math problems was recorded for each subject.  

Control Group

Sleep Deprivation

Food Deprivation

1

2

3

4

7

8

9

10

4

5

6

7

The group means and variances can be shown below.

 

 

The Overall F Test vs. A Priori Comparisons

1. Should an overall F test be called for?

No. The overall F test tests the significance of all the differences among group means taken together. 

2. A Priori Comparisons

Tests between sample means that are planned before collecting data are called a priori comparisons or planned comparisons. A priori comparisons are tested directly regardless of the statistical significance of the overall F test.  

Planned Comparisons Using Contrasts  

The contrast coefficients are chosen to test the corresponding hypotheses of interest.

1. Contrast 1

Groups being compared have either positive or negative coefficients, while groups not being compared have coefficients equal to zero. Also, the individual coefficients should sum to zero.

Thus, the first set of contrast coefficients is [1, -1, 0].

Group Control (1) Sleep Deprivation (2) Food Deprivation (3)
Coefficient 1 -1 0

The comparison value is computed as (1)M1 + (-1)M2 + (0)M3 = M1 - M2

Group Control (1) Sleep Deprivation (2) Food Deprivation (3)
Coefficient 1 -1 0
Mean 2.5 8.5 5.5

          M1 - M2 = 2.5 - 8.5 = -6

 2. Contrast 2

Group Control (1) Sleep Deprivation (2) Food Deprivation (3)
Coefficient 1 0 -1

M1 - M3 = 2.5 - 5.5 = -3

3. Hypotheses

Contrast 1

The null hypothesis is



Thus,

The alternative hypothesis is



Thus,


 

Use a .05 significant level.

Contrast 2

The null hypothesis is



  Thus,


 

The alternative hypothesis is



Thus,

Use a .05 significant level.
 

4. Control Type I Error 

No correction

Planned comparisons are usually evaluated at an uncorrected significance level. Note that there are two planned comparisons being made. The number of the planned comparisons does not exceed the degrees of freedom associated with the treatment effects (There are three groups and the degrees of freedom are equal to 3-1=2). 
 


SPSS for Windows

Enter Data. 

 

A. From the menus choose: Analyze \ Compare Means \ One-Way ANOVA.

Select the dependent variable (error) and the factor variable (group).

B. To specify a priori contrasts, click on the Contrasts button in the One-Way ANOVA dialog box.

1. Click inside the Coefficients text box. Enter a coefficient value for each group and click on Add after each entry. The first set of contrast coefficients is: 1   -1   0

2. To specify the second contrast, click on the Next button.

Enter a coefficient value for each group and click on Add after each entry. Recall that the second set of contrast coefficients is: 1   0   -1

         Click Continue.  

C. To obtain descriptive statistics and the Levene statistic, click on Options. Select Descriptive and Homogeneity-of-variance. Click Continue. Click OK.

 

SPSS Printout  

1. Descriptive Statistics

Observe the measures of central tendency and variability. Examine the 95% confidence intervals for three groups.


2. Test of Homogeneity of Variance

   

Is the assumption of homogeneity of variances met?


3. ANOVA Table

What do you observe?


4. Contrast Coefficients


5. Contrast Tests


CONTRAST 1

the control group against the sleep deprivation group (contrast coefficients 1, -1, 0)  
 

Group Control (1) Sleep Deprivation (2) Food Deprivation (3)
Coefficient 1 -1 0
Mean 2.5 8.5 5.5

 

A. Hypotheses  

The null hypothesis is



The alternative hypothesis is

Use a .05 significant level.

B. The t-Test

a. Calculate the value of contrast (the sum of weighted means).

Apply the first set of coefficients to the sample means to obtain the comparison value. 

(1)M1 + (-1)M2 + (0)M3 = (1)M1 + (-1)M2 = 2.5 - 8.5 = -6

b. Calculate the estimated (unbiased) standard error of comparison.  

 

Notice that there are 4 subjects in each group  

c. Form a t ratio. t = value of contrast / standard error of the comparison

t = (-6) / .9129 = -6.573

d. For a p-value of .000, report it as p < .001. The observed significance level is less than .05. Contrast 1 is significant, t(9) = -6.573, p< .05.  There is a significant difference between the sleep deprivation group and the control group.
 

CONTRAST 2

the control group against the food deprivation group (contrast coefficients 1, 0, -1)  
 

Group Control (1) Sleep Deprivation (2) Food Deprivation (3)
Coefficient 1 0 -1
Mean 2.5 8.5 5.5

 

A. Hypotheses  

The null hypothesis is

The alternative hypothesis is

Use a .05 significant level.

B. The t Test

a. Calculate the value of contrast.  

(1)M1 + (0)M2 + (-1)M3 = M1 - M3 = 2.5 - 5.5 = -3.

b. Calculate the unbiased standard error of comparison.

 

Find the value of MSw from the ANOVA table. MSw equals 1.667.
Notice that there are 4 subjects in each group  

c. Form a t ratio. t = value of contrast / standard error of the comparison

t = -3 / .9129 = -3.286

d. The observed significance level is .009. Contrast 2 is significant, t(9) = -3.286, p < .05. There is a significant difference between the food deprivation group and the control group.