Chapter 13 SPSS Analysis of CRAC-4 Design (Two Covariates)

Chapter 13 SPSS Analysis of CRAC-4 Design (Two Covariates)

Data from Table 13.6-1

Exploratory Data Analysis With SPSS

Prior to performing an analysis of covariance, an exploratory data analysis should be performed. The exploratory data analysis may uncover data recording errors, ANOVA assumptions that appear untenable, and unexpected promising lines of investigation. The data for the four methods of teaching arithmetic in Table 13.6-1 of Experimental Design: Procedures for the Behavioral Sciences (page 643) are used to illustrate the procedures for a completely randomized analysis of covariance design with two covariates. Treatment A is method of teaching arithmetic. The covariates are a measure of a student’s intelligence (X) and initial arithmetic knowledge (Z). The dependent variable (Y) is the student’s arithmetic achievement at the conclusion of the course. The data are as follows.

Table 13.6-1. Teaching-Methods Data

Treatment (Y) and Covariates (X) and (Z) Levels

ay1 ax1 az1 ay2 ax2 az2 ay3 ax3 az3 ay4 ax4 az4

3 42 3 4 47 4 7 61 5 7 65 2 6 57 5 5 49 6 8 65 7 8 74 4 3 33 4 4 42 5 7 64 5 9 80 5 3 47 4 3 41 2 6 56 4 8 73 5 1 32 0 2 38 1 5 52 2 10 85 6 2 35 1 3 43 2 6 58 3 10 82 6 2 33 0 4 48 5 5 53 3 9 78 5 2 39 2 3 45 3 6 54 4 11 89 7

1. Double click on the SPSS icon to open SPSS. This action opens the window shown below that ask, What would you like to do?

2 2. Select the Type in data button and then the OK button in the lower right corner.

Type in data!OK This action opens the SPSS Statistics Data Editor window shown next. At the bottom lower left of the

window are two rectangular buttons: Data View and Variable View. The Data View button is highlighted, which means that the window for entering new data is open. Before entering new data, the names and details of the variables in the new data set should be defined.

3. Click on the Variable View button to open the following SPSS Statistics Data Editor Variable

View window where the characteristics of the data are defined.

You can go directly to the Variable View window. Instead of clicking on the OK button in the first window, click on the Cancel button in the lower right corner of the first window.

4. Use row 1 of the Variable View window to describe the characteristics of the subject numbers. Use row

2 to describe treatment A, method of teaching aritmetic. Use row 3 to describe the first covariate, intelligence of the students, and row 4 to describe the second covariate, initial arithmetic knowledge.

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Use row 5 to describe the dependent variable, arithmetic achievement. For the data in Table 13.6-1, fill in the columns of row 1 as follows:

Columns 1. Name. Name the variable Subject. A variable name can have no more than 64 characters. The first character must be an upper or lower case letter; the remaining characters can be any letter, digit, or the symbols @, #, _, or $. No spaces can appear in the name. Also, variable names cannot end with a period.

Column 2. Type. This column enables you to define the variable type. The default type is Numeric for number. You can change the variable type by clicking on the Type cell. This action opens the Variable Type window where you can select from nine variable types including Scientific notation, Date, Custom currency, String, and Text. If you have changed the default, you need to click on the OK button at the bottom of the Variable Type window to return to the SPSS Statistics Data Editor window.

Column 3. Width. This column enables you to define the number of characters that are shown for a variable in the SPSS Statistics Data Editor. The default is 8 characters. When you click on the Width cell, a blue scroll box appears on the right side of the cell where the options are 1, . . . , 40 characters.

Column 4. Decimals. This column enables you to define the number of characters to the right of the decimal point. When you click on the Decimal cell, a blue scroll box appears on the right side of the cell. The options are 0, . . . , 16 decimals: scroll to 0.

Column 5. Label. This column enables you to provide a descriptive label for the subject: type in Subject.

Column 6. Values. This column is used with grouping variables. Do not change None, which is the default.

Column 7. Missing. Do not change None, which is the default. The data in Table 13.6-1 have no missing values.

When you click on the Missing values cell, a blue button appears on the right side of the cell. If you click on the blue button, the Missing Values window that is shown next opens. This window enables you to identify discrete missing values or a range of missing values.

Click on the OK button at the bottom of the window to return to the SPSS Statistics Data Editor window.

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Column 8. Columns. This column enables you to specify the number of characters in a column. When you click on the Columns cell, a blue scroll box appears on the right side of the cell where you can select from 1 to 255 characters. Do not change 8, which is the default.

Column 9. Align. This column enables you to determine the alignment of the columns. When you click on the Align cell, the cell enlarges and provides three options: Left, Right, and Center. Do not change Right, which is the default.

Column 10. Measure. This column enables you to specify the level of measurement of your variable. When you click on the Measure cell, the cell enlarges and provides three options: Scale, Ordinal, and Nominal. Click on the Nominal option. Scale denotes either interval or ratio measurement.

Column 11. Role. This column enables you to specify the role that the variable plays in the data set such as input, output, or partitioning data into samples. Do not change Input, which is the default. When you click on the Role cell, the cell enlarges and provides six options: Input, Target, Both, None, Partition, and Split. Further details about Role are available in SPSS’s help menu (Help Topics roles).

5. After you have entered the information for the subject in row 1, it is a good idea to give your SPSS file a name and save the file. Select File in the menu and then Save as to name the file Table 13.6-1. The SPSS Statistics Data Editor Variable View window should appear as follows. Notice that Untitled1 in the top left of the window has been replaced with Table 13.6-1.sav.

6. Use row two of the Variable View window to describe the characteristics of treatment A, method of

teaching arithmetic. For the data in Table 13.6-1, fill in the columns of row 2 as follows.

Column 1. Name. Name the variable Teach_method.

Column 2. Type. Do not change Numeric, which is the default.

Column 3. Do not change 8, which is the default.

Column 4. Decimals. Click on the Decimal cell. In the blue scroll box on the right side of the cell, scroll to 0.

Column 5. Label. Label the independent variable Teaching method.

! !

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Column 6. Values. This column enables you to provide labels for various levels of treatment A. When you click in the Values cell, a blue button appears on the right side of the cell. Click on the blue button to open the Value Labels window that is shown next.

Type “1” in the Value rectangle and “ay1” in the Label rectangle. Then click on the Add button to move the label into the large rectangle. Repeat the procedure by entering “2” and “ay2,” “3” and “ay3,” and “4” and “ay4.” The Value Labels window changes as shown next. Click on OK to return to the Variables View window.

Column 7. Missing. Do not change None, which is the default.

Column 8. Columns. Do not change 8, which is the default.

Column 9. Align. Do not change Right, which is the default.

Column 10. Measure. Do not change nominal, which is the default.

Column 11. Role. Do not change Input; which is the default.

After you have described the characteristics of the independent variable, the Variable View window should appear as follows.

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7. Use row 3 of the Variable View window to describe the characteristics of the first covariate,

intelligence. For the data in Table 13.6-1, fill in the columns of row 3 as follows.

Column 1. Name. Name the variable intelligence.




Column 5. Label. Label the first covariate Intelligence.

Column 6. Values. Do not change None, which is the default.




Column 10. Measure. Click on the Measure cell and select the Scale option.


After you have described the characteristics of the first covariate, the Variable View window should appear as shown next.

7 8. Use row 4 of the Variable View window to describe the characteristics of the second covariate,

arithmetic knowledge. For the data in Table 13.6-1, fill in the columns of row 4 as follows.

Column 1. Name. Name the variable Arith_know.




Column 5. Label. Label the independent variable Arith knowledge.







After you have described the characteristics of the second covariate, the Variable View window should appear as shown next.

9. Use row 5 of the Variable View window to describe the characteristics of the dependent variable, arithmetic achievement of the students. For the data in Table 13.6-1, fill in the columns of row 5 as follows.

Column 1. Name. Name the variable Arith_ach.




Column 5. Label. Label the dependent variable Arithmetic achievement.



8





After you have described the characteristics of the dependent variable, the Variable View window should appear as shown next.

10. Click on Data View in the lower left of the SPSS Statistics Data Editor window to open the Data

View window. Enter the subject number (1–32), teaching method (1–4), values for the two covariates, and arithmetic achievement score as shown next.

9 11. To obtain descriptive statistics for the four methods of teaching arithmetic, click on Analyze in the

menu. Select Descriptive Statistics from the drop-down menu and then Explore (Analyze!Descriptive Statistics!Explore). These actions open the Explore window shown next where the independent and dependent variables are identified.

Select Teaching method and then click on the arrow beside Factor List to move Teaching method into the Factor List rectangle. Select Arithmetic achievement and then click the arrow beside Dependent List to move Arithmetic achievement into the Dependent List rectangle.

After these actions, the Explore window should appear as shown next.

12. Click on the Statistics button in the upper right corner of the Explore window to open the Explore:

Statistics window shown next. The Descriptives box is checked and 95 appears in the Confidence Interval for Mean box. Other confidence interval coefficients such as 90, 92.5, and 99 can be inserted in the box.

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Click on the Continue button at the bottom of the window to return to the Explore window.

13. Click on the Plots button in the upper right corner of the Explore window to open the Explore: Plots

window shown next. The Factor levels together button and the Stem-and-leaf box are selected. Click on the Stem-and-leaf box to unselect it. Click on the Histogram box to select it.

11 14. Click on the Continue button at the bottom of the Explore: Plots window to return to the Explore

window. Click on the OK button at the bottom of the Explore window to obtain the following descriptive statistics for the hand-steadiness data.

Results of the Exploratory Data Analysis

This output summarizes the information that was processed for the four teaching methods. It is important to examine the table to be sure that SPSS has correctly interpreted your instructions about the variables and number of observations.

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The means shown here are the unadjusted means. The estimators of the population variances are similar. The largest variance, treatment level a1, is only 2.214/0.857 = 2.6 times larger than the smallest treatment variance, treatment level a2. The size of the variances is not related to the size of the unadjusted sample means. The sample means, trimmed means, and medians are similar. This suggests that the distributions are fairly symmetrical. The symmetry of the distributions also is supported by the histograms and box plots shown next.

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The box plots suggest that the populations are reasonably symmetrical. The box plots identified one outliers: observation 2 in treatment level a1. The data collection procedures should be examined to determine if reasons can be found to question the accuracy of this outliers. I assume, for the purposes of this example, that the outlier is simply a manifestation of the random variability among the observations.

Test of Homogeneity of Regression Slopes for Intelligence 1. To test the ANCOVA assumption of homogeneity of regression slopes for intelligence, click on Analyze in

the Menu Bar. Select General Linear Model from the pull-down menu and then Univariate (Analyze

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General Linear Model Univariate). These selections open the Univariate window that is shown next in which you specify which variable is the independent variable, which is the dependent variable, and which is the covariate.

2. Select Teaching method and click on the arrow beside the Fixed Factor(s) box. The independent variable, Teaching method, will appear in the Fixed Factor(s) box. Select Intelligence and click on the arrow beside the Covariate(s) box. Select Arithmetic achievement and click on the arrow beside the Dependent Variable box. After these actions, the Univariate window changes as shown next.

! !

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3. Click on the Model button in the Univariate window to open the Univariate: Model window.

4. Click on the Custom button to define a model that contains the main effects Teach_method and Intelligence and an interaction term Teach_method × Intelligence. Click on the drop-down menu in the center of the window to change Interaction to Main Effects. Select Teach_method and click on the arrow to move it into the Model box. Repeat the procedure for Intelligence. Click on the drop-down menu to change Main Effects to Interaction. Use “control” + “command” to select Teach_method and Intelligence simultaneously in the Factors & Covariates rectangle. Click on the arrow to move them into the Model box. The Univariate Model window should appear as shown next.

Click on the Continue button to return to the Univariate window. Then click on OK in the Univariate window to obtain the following output.

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A test of the ANCOVA assumption of homogeneity of the population regression slopes is given by F = Teach_method*Intelligence/Error = 0.144/0.276 = 0.552 (p =.671). When, as in this example, n ≤ 8 it is customary to adopt the α = .25 level of significance in performing a preliminary test on the model. The adoption of α = .25 results in a more powerful test of the model assumption. Because p =.671 > .25, the assumption of homogeneity of the population regression slopes is tenable.

Test of Homogeneity of Regression Slopes for Arithmetic Knowledge 1. To test the ANCOVA assumption of homogeneity of regression slopes for initial arithmetic knowledge,

click on Analyze in the Menu Bar. Select General Linear Model from the pull-down menu and then Univariate (Analyze General Linear Model Univariate). These selections open the Univariate window that is shown next in which you specify which variable is the independent variable, which is the dependent variable, and which is the covariate.

! !

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2. Select Teaching method and click on the arrow beside the Fixed Factor(s) box. The independent

variable, Teaching method, will appear in the Fixed Factor(s) box. Select Arith knowledge and click on the arrow beside the Covariate(s) box. Select Arithmetic achievement and click on the arrow beside the Dependent Variable box. After these actions, the Univariate window changes as shown next.

3. Click on the Model button in the Univariate window to open the Univariate: Model window.

4. Click on the Custom button to define a model that contains the main effects Teach_method and Arith_know and an interaction term Teach_method × Arith_know. Click on the drop-down menu in the center of the window to change Interaction to Main Effects. Select Teach_method and click

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on the arrow to move it into the Model box. Repeat the procedure for Arith_know. Click on the drop-down menu to change Main Effects to Interaction. Use “control” + “command” to select Teach_method and Arith_know simultaneously in the Factors & Covariates rectangle. Click on the arrow to move them into the Model box. The Univariate Model window should appear as shown next.

Click on the Continue button to return to the Univariate window. Then click on OK in the Univariate window to obtain the following output.

A test of the ANCOVA assumption of homogeneity of the population regression slopes is given by F = Teach_method*Arith_know/Error = 0.303/0.302 = 1.003 (p =.409). When, as in this example, n ≤ 8 it is customary to adopt the α = .25 level of significance in performing a preliminary test on the model. The adoption of α = .25 results in a more powerful test of the model assumption. Because p =.409 > .25, the assumption of homogeneity of the population regression slopes is tenable.

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Analysis of Variance With SPSS

1. To perform an analysis of covariance on the four methods of teaching arithmetic, click on Analyze in the menu bar. Select General Linear Model from the pull-down menu and then Univariate (AnalyzeGeneral Linear Model Univariate). These selections open the following Univariate window.

2. Select Teaching method and click on the arrow beside the Fixed Factor(s) box. The independent variable, Teaching method, will appear in the Fixed Factor(s) box. Select Intelligence and click on the arrow beside the Covariate(s) box. Select Arith knowledge and click on the arrow beside the Covariate(s) box. Select Arithmetic achievement and click on the arrow beside the Dependent Variable box. After these actions, the Univariate window changes as shown next.

!

!

20 3. Notice that the Post Hoc button on the right side of the Univariate window is not available. There are

several ways to obtain multiple comparisons. One way is to click on the Contrasts button on the right side of the Univariate window. This action opens the following Univariate: Contrasts window.

Click on the Contrast scroll rectangle to select one of following six types of contrasts.

Deviation: Each mean is compared with the grand mean. Simple: A pre-specified Reference Category mean (First or Last) is compared with each of the other means. Difference (Reverse Helmert contrast): Compares the mean of each level (except the first) to the mean of previous levels. Helmert: Starting from the leftmost mean in an array, each mean is compared with the mean of the remaining means. Repeated: The first mean is compared with the second, the second with the third, third with the fourth, and so on. Polynomial: Compares the linear effect, quadratic effect, cubic effect, and so on. The first degree of freedom contains the linear effect across all categories; the second degree of freedom, the quadratic effect; and so on.

After a contrast type has been selected, click on Change to complete the specification. Click on Continue to return to the Univariate window. In this example, none of these contrasts are particularly interesting.

4. A second way to obtain post hoc tests is to click on the Options button on the right side of the Univariate

window. This action opens the following Univariate: Options window.

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To specify post hoc tests, select the independent variable, Teach_method and click on the arrow beside

the Display Means for rectangle. Click on the Compare main effects box. The scroll box Confidence interval adjustment becomes active. Click on the scroll box and select Sidak. In the Display panel, click on Descriptive statistics, Estimates of effect size, and Homogeneity tests.

Click on the Continue button at the bottom of the Univariate: Options window to return to the Univariate window. Click on the OK button at the bottom of the Univariate window to obtain the results of the completely randomized ANCOVA.

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Results of the Analysis of Covariance

SPSS lists of all of the results sub tables on the left side of the results window. A list of these sub tables is shown below.

This output summarizes the information that was processed for Treatment A.

The means shown here are the unadjusted means.

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Levene’s test provides a test of the ANCOVA assumption of the homogeneity of the error variances for the four treatment populations. In order to increase the power of the test, it is a common practice to adopt the α = .15 or .25 level of significance. As I discuss in Section 3.5 of Experimental Design: Procedures for the Behavioral Sciences, Levene’s test is less robust to nonnormality than the Brown-Forsythe test. The computation of the latter test is illustrated on pages 102–103 of Experimental Design: Procedures for the Behavioral Sciences.

After adjusting for the two concomitant variables, the omnibus null hypothesis for the teaching methods data was rejected: F = MSBG/MSWG = 0.884/0.148 = 5.955, p = .003.

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The means in this table have been adjusted for the concomitant variables and should be used for performing multiple comparisons.

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The omnibus null hypothesis for the teaching methods data was rejected: F = MSBG/MSWG = 0.884/0.148 = 5.955, p = .003. The sums of squares, mean squares, and F statistic differ slightly from the values on pages 644645 of Experimental Design: Procedures for the Behavioral Sciences. The differences are due to rounding error. All of the intermediate computations on pages 644–645 are rounded to three decimal places. If the computations are carried to eight decimal places, the values are the same as the SPSS values.

Reference

Kirk, R. E. (2013). Experimental Design: Procedures for the Behavioral Sciences. (4th ed.). Thousand Oaks, CA: Sage.

Suggested Readings

Brace, N, Kemp, R., & Snelgar, R. (2013). SPSS for Psychologists (5th ed.) New York, NY: Routledge.

Kinnear, P. R., & Gray, C. D. (2011). IBM SPSS Statistics Made Simple 18. New York, NY: Psychology Press.

Chapter 13 SPSS Analysis of CRAC-4 Design (Two Covariates)

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Transcript of Chapter 13 SPSS Analysis of CRAC-4 Design (Two Covariates)