CROSSED DESIGNS

As discussed earlier, a crossed design occurs when each subject sees each treatment level, that is, when there is more than one score per subject. The purpose of the analysis is to determine if the effects of the treatment are real, or greater than expected by chance alone.

EXAMPLE DESIGN

An experimenter is interested in the difference of finger-tapping speed by the right and left hands. She believes that if a difference is found, it will confirm a theory about hemispheric differences (left vs. right) in the brain.

A sample of thirteen subjects (N=13) is taken from a population of adults. Six subjects tap for fifteen seconds with their right-hand ring finger. Seven subjects tapped with their left hand. After the number of taps have been recorded, the subjects tap again, but with the opposite hand. Thus each subject taps with both the right hand and the left hand. They appeared in each level of the treatment condition.

RAW SCORES

After the data is collected, it is usually arranged in a table like the following:

Note that the two scores for each subject are written in the same column.

STEP ONE - FIND THE DIFFERENCE SCORES

In analysis of crossed designs, first calculate the difference scores for each subject. These scores become the basic unit of analysis. For example, finding difference scores from the data presented above would result in the following table:

The difference scores will be symbolized by D_i to differentiate them from raw scores, symbolized by X_i.

STEP TWO - FIND THE MEAN AND STANDARD DEVIATION OF D_i

The next step is to enter the difference scores into the calculator and calculate the mean and standard deviation. For example, the mean and standard deviation of the difference scores presented above are:

Difference

Mean = = 4.69

Standard Deviation = s_D = 7.146

The mean and standard deviation of the difference scores will be represented by and S_D, respectively.

THE MODEL UNDER THE NULL HYPOTHESIS

The Null Hypothesis states that there are no effects. That is, if the experiment was conducted with an infinite number of subjects, the average difference between the right and left hand would be zero (=0).

If the experiment using thirteen subjects was repeated an infinite number of times assuming the Null Hypothesis was true, then a model could be created of the means of these experiments. This model would be the sampling distribution of the mean difference scores. The central limit theorem states that the sampling distribution of the mean would have a mean equal to the mean of the population model. In this case the mean would equal 0.0, and a standard error represented by . The standard error could be computed by the following formula.

The only difficulty in this case is that the standard deviation of the population model, is not known.

The standard deviation of the population model can be estimated, however, using the sample standard deviation, s_D. This estimation adds error to the procedure and requires the use of the t-distribution rather than the normal curve. The t-distribution will be discussed in greater detail in a later chapter.

STEP THREE - ESTIMATE THE STANDARD ERROR

The standard error of the mean difference score, , is estimated by , which is calculated by the following formula:

Using this formula on the example data yields:

STEP FOUR - CALCULATE t_OBS

The obtained mean difference is transformed into a z-score relative to the model under the Null Hypothesis using the following formula:

In the example data the preceding equation produces the following:

STEP FIVE - FIND t_CRIT

The critical value for t_OBS, called t_CRIT, is found in the t tables. The degrees of freedom must first be calculated using the following formula:

In this case, it results in the following:

Finding the critical value of t using 12 degrees of freedom, a two-tailed test, and the .05 level of significance yields:

Since the observed value of t_obs = 2.366 is greater than the critical value (2.179), the model under the Null Hypothesis is rejected and the hypothesis of real effects accepted. The mean difference is significant. In this case the conclusion would be made that the right hand taps faster than the left hand. This is because the mean difference is greater than zero, meaning that the top number is larger.

USING SPSS TO COMPUTE A CROSSED T-TEST

The data is entered into an SPSS data file with each subject as a row and the two variables for each subject as a column. In the example data, there would be thirteen rows and two columns, one each for the right and left hand data. The data file appears as follows:

What I call a crossed t-test, the SPSS package calls a paired-samples T Test. This statistical procedure is accessed as follows:

The user must then tell SPSS which variables are to be compared. The example data analysis would proceed as follows:

The output from this procedure would include means

and t-test results.

Note that the results from using SPSS are within rounding error of the results computed by using the formulas and a hand-held calculator.

CONCLUSION

When each subject sees each level of treatment, a crossed design results. In a crossed design, the difference scores between the two treatment levels are first calculated for each subject. Next, the mean and standard deviation of the difference scores are found. Following this, the standard error of the mean difference scores is estimated and the t observed value found. The t observed is then compared with the critical value obtained from a table in the back of the book. If the t observed is larger than the t critical value, the Null Hypothesis is rejected and the hypothesis of real effects is accepted. In the opposite case, where the t observed is less than the t critical value, the Null Hypothesis is retained with no conclusion about the reality of effects.