# Mixed 2 x 3 ANOVA. Notes

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1 Mixed 2 x 3 ANOVA This section explains how to perform an ANOVA when one of the variables takes the form of repeated measures and the other variable is between-subjects that is, independent groups of participants are identifiable. Here three repeated measures are taken for each participant and there are two separate groups of participants. In the example now provided, there are two independent groups: (1) university modern language students studying German and (2) university modern language students studying French. (In other words, here what is involved is a classification variable). 1 The repeated measures aspect is the fact that the students were given a test of fluency in their target language at the end of each of their three years of study. A scale 0 20 was used in the test of fluency. The test, of course, was standardised across the two languages / groups. The (fictitious) results of this investigation into second language fluency are presented in the table on page 137. There were 6 students of German and 5 students of French. The abbreviation P is used for Participant. Notes In this example the two independent groups are based purely upon the classification of each participant: (1) students of German and (2) students of French. In some research, instead of basing the groups upon classification in this manner, participants are (for example) randomly allocated to two (independent) groups. Two such experimental groups might be, for example, where group (A) are subjected to one teaching style (throughout their three years of study) and group (B) are subjected to a quite different style of instruction. The repeated measures aspect would be the test on target language fluency at the end of each of the three years of study. 1 That is to say that any particular student can be classed as either being a student of German or as a student of French. Mixed 2 x 3 ANOVA Page 131

2 In this example, as elsewhere in this workbook, the results (data) provided are based on rather low Ns (number of participants). For the ANOVA to be sufficiently powerful, you would normally need rather more participants. (Usually you should aim for at least ten participants in each experimental / classification group). Low numbers of participants are used here simply for ease of presentation. If you already know how to code variables and how to enter the data for a Mixed 2 x 3 ANOVA, just load the following data set and then move on to the instructions that explain how to perform the Mixed 2 x 3 ANOVA itself. This data set is available at the supporting website for this workbook: Data set: Mixed 2 x 3 ANOVA (Workbook Example) Code to access the data sets from the website: 2211 Otherwise, follow the instructions regarding data entry. Mixed 2 x 3 ANOVA Page 132

3 Students of German Students of French Scores at the end of Year ONE (P1) 2 (P2) 4 (P3) 6 (P4) 6 (P5) 3 (P6) 4 Mean = 4.17 (P7) 5 (P8) 4 (P9) 8 (P10) 7 (P11) 5 Mean = 5.80 Mean of means for Year ONE = 4.98 Scores at the end of Year TWO (P1) 4 (P2) 7 (P3) 8 (P4) 9 (P5) 5 (P6) 6 Mean = 6.50 (P7) 10 (P8) 11 (P9) 14 (P10) 14 (P11) 15 Mean = Mean of means for Year TWO = 9.60 Scores at the end of Year THREE (P1) 8 (P2) 8 (P3) 10 (P4) 12 (P5) 7 (P6) 9 Mean = 9.00 (P7) 15 (P8) 17 (P9) 18 (P10) 19 (P11) 19 Mean = Mean of means for Year THREE = Mean of means for German = 6.56 Mean of means for French = Mixed 2 x 3 ANOVA Page 133

4 Load IBM SPSS Statistics Click on circle to Type in Data OK Click on Variable View Enter (top-left under Name) the word Language Under Label, enter the word Language Click into the Values box and then click on [ ] Click into the Value box and enter the digit 1 Click into the Label box and enter the word German Add Click into the Value box and enter the digit 2 Click onto the Label box and enter the word French Add OK Mixed 2 x 3 ANOVA Page 134

5 Enter (top/left) in the box under the word language, the word One Under Label, enter the words fluency at the end of year one. Enter (top/left) in the box under the word One, the word Two Under Label, enter the words fluency at the end of year two. Enter (top/left) in the box under the word Two, the word Three Under Label, enter the words fluency at the end of year three. Your screen should now look as indicated below. Click on Data View [bottom/left of your screen] As six students of German were tested in this investigation, enter in the first column the digit 1 [enter] do this six times. This followed by the digit 2 [enter] five times as five students of French were tested. Then enter each participant s score as appropriate. When all these data have been entered, your screen should look as indicated below. Mixed 2 x 3 ANOVA Page 135

6 Analyse General Linear Model Repeated measures Within subjects factor. Here delete the words factor1 and type in the word year. The word year is suggested here because the students were given the test of fluency in the target language at the end of each of their three years of study. In other words within each of the groups, this is the measure that was repeatedly (i.e. three times) taken. In the box with the words Number of Levels, type in the digit 3 Add Click on Define Highlight in yellow the following three variables: Mixed 2 x 3 ANOVA Page 136

7 fluency at the end of year one fluency at the end of year two fluency at the end of year three With these highlighted, click these over to the right Highlight the word Language Click over to the box which defines the Between-Subjects Factor(s) Click on Plots Click Year over to the horizontal axis Click Language to separate lines Add Continue Mixed 2 x 3 ANOVA Page 137

8 Click on Options Highlight the following in yellow: language year language * year With these highlighted, click these over to the box labelled Display Means for: Select (by ticking the box) Descriptive statistics Continue OK You should now be presented with the IBM SPSS Statistics (18) output. Selected sections of this output are now provided and interpreted / commented upon. The above examines possible within-subjects effects : Mixed 2 x 3 ANOVA Page 138

9 The F-value of is significant at p <.001. This means that, ignoring whether participants were studying German or French, there is an overall significant difference in fluency between the three years of study. This is referred to as a main effect for year. The right-hand column in the data table (page 136) clearly reflects / represents this fact. The F-value of is also significant at p <.001. This means that there is a significant interaction between the two independent variables i.e. target language and year of study. If you examine the graph on the next page, it is clear to see that as students progress through the three years, then the gap between fluency abilities widens: students of French increasingly out-performed the students of German as the three years pass. Measure:MEASURE_1 Transformed Variable:Average Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Intercept Language Error The above section of the IBM SPSS Statistics output indicates a significant main effect for Language : F = 30.48; df = 1,9; p <.001. This means that in general / overall students of French are significantly more fluent than students of German. The very bottom line (row) of the data table (page 136) clearly represents this fact. Mixed 2 x 3 ANOVA Page 139

10 Summary The ANOVA analysis has revealed that the two possible main effects have been shown to be significant. Students get progressively more fluent as they progress through their studies this is the case whether they are studying German or French. Students of French are generally more fluent than students of German. The ANOVA analysis has also revealed that there is a significant interaction between the two independent variables. You should be aware that when a significant interaction is obtained the interpretation of findings should be based primarily on the interaction. We can begin to interpret the interaction by making reference to the graph shown below. (The top line connects the three means for the students of French. The bottom line connects the three means for the students of German). Mixed 2 x 3 ANOVA Page 140

11 Further examination of the data (1) You can perform further statistical analysis to investigate the interaction in more detail. For example you may interrogate the interaction to determine if the students of French are significantly more fluent than students of German after each year of study. This could be done by employing three separate Independent t-tests: that is, one for each year of study. In order to conduct the t-tests Analyse Compare means Independent-Samples t-test Highlight in blue the following three variables: fluency at the end of year one fluency at the end of year two fluency at the end of year three With these highlighted, click these over to the right into the Test Variable(s) box. Highlight the word Language. Click over to the Grouping Variable box. Click the Define Groups button Enter 1 for Group 1 and 2 for Group 2 Continue OK The IBM SPSS Statistics output should now include the results of these three t- Tests: Mixed 2 x 3 ANOVA Page 141

12 The above analysis would be reported thus in a journal report: No significant difference was observed between the two groups of language students at the end of their first year of study. At the end of the second year, students of French scored higher on the test of fluency in their target language (M = 12.80, SD = 2.17) than students of German (M = 6.50, SD = 1.87). A significant difference between the two groups was observed, t(9) = 5.18, p =.001. At the end of the third year, the gap between the two groups widened: students of French scored higher on the test of fluency in their target language (M = 17.60, SD = 1.67) than students of German (M = 9.00, SD = 1.79). A significant difference between the two groups was observed, t(9) = 8.17, p <.001. Note: the values of t presented in the IBM SPSS Statistics output have negative values (-5.18 and -8.17). The minus sign is not employed in the above statements because it is the numerical value that is important. Had the groups been entered into IBM SPSS Statistics the other way around (with students of French coded as 1 and students of German coded as 2) then the values of t would have had a positive value. The result of Levene s Test for Equality of Variances (presented in the IBM SPSS output) is not significant. Hence the values of t reported above are the values for Equal variances assumed. Mixed 2 x 3 ANOVA Page 142

13 (2) Alternatively you may wish to see if each of the groups of students have improved significantly during the three years that they have studied their respective languages. To do this you would need to employ two separate oneway repeated measures ANOVAs, one ANOVA for students of French and a separate ANOVA for the students of German. 2 In order to instruct IBM SPSS Statistics to perform the separate repeated measures ANOVAs you will need to employ a data manipulation command. With the Data View screen showing, click on [Data] from the menu at the top of the screen Select Cases Click in the circle next to If condition is satisfied Click in the box beneath [If] Highlight in yellow Language and click this over to the box to the right Click on = Click on 1 You screen should look like this: 2 If the one-way repeated measures ANOVAs indicate significance, a series of related t-tests should then also be performed in order to tease out exactly which of the measures differ significantly from which. Mixed 2 x 3 ANOVA Page 143

14 Continue OK You will have succeeded in selecting only the six students of German. You can now perform the suggested one-way repeated measures ANOVA on the data for the students of German. If you go back to Data View, the screen should look like this: Mixed 2 x 3 ANOVA Page 144

15 You can now perform a one-way repeated measures ANOVA to see if the students of German have improved significantly during their three years of study. The above process should now be repeated. But this time, select only the students of French. Mixed 2 x 3 ANOVA Page 145

16 Workshop Exercise (12): Mixed 2 x 3 ANOVA A longitudinal study was conducted in which children were tested for mathematical ability when they were four years old; three years later, when they were seven years old; and then when they were ten years old. 300 boys and 300 girls were tested. Hence the total number of participants was 600. Conduct the appropriate ANOVA on these data. The IBM SPSS Statistics data set can be loaded from the following data file. This data set is available at the supporting website for this workbook: 2 x 3 Mixed ANOVA (WORKSHOP Exercise 12) Code to access the data sets from the website: 2211 Note Appendix II of this workbook provides a model Results section relating to this example. This is written in accordance with the typical requirements (regarding format and presentation) for journal reports. Mixed 2 x 3 ANOVA Page 146

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