# Factorial ANOVA and ANCOVA Output Application in Criminology

## Introduction

A researcher was interested in finding the best way to educate elementary-age children in mathematics. In particular, she hypothesized that fifth-grade girls do better in small class sizes while boys excel in larger classes. The researcher performed a pilot study throughout the school district where classroom sizes were reduced into three groups as follows: small classrooms containing no more than 10 children, medium classrooms having between 11 and 19 children, and large classrooms having 20 or more children. This paper is a write-up of the exploratory data analysis for the variables used in the study using SPSS. In addition, the paper presents a factorial ANOVA of the dataset as well as a hypothetical ANCOVA output for predicting criminal behavior.

## Exploratory Data Analysis

The mean math score for girls studying in a class of 10 or fewer students was 93.80 with a standard deviation of 3.938. The minimum math score for this group was 88.00 whereas the maximum score was 98.00. The mean math score for boys in a class of 10 or fewer students was 92.70 with a standard deviation of 3.434. The minimum math score for this group was 87.00 whereas the maximum score was 99.00 (Table 1). It is therefore evident that girls performed slightly better in math than boys when studying a class of 10 or fewer students.

In a class of 11 to 19 children, the mean math score for girls was 88.50 with a standard deviation of 3.979. In the same class size, the mean math score for boys was 89.70 with a standard deviation of 2.406. The minimum score for girls was 82.00 whereas the minimum score for boys was 86.00. The maximum score on the other hand was 95.00 for girls and 93.00 for boys (Table 2). The mean math scores for this group, therefore, indicate that math performance for boys was better than that of girls.

The mean math score for girls in a class of 20 or more students was 79.20 with a standard deviation of 4.184 whereas the mean math score for boys in the same class size was 91.20 with a standard deviation of 3.225. The highest math score for girls in this group was 86.00 whereas the highest score for boys was 98.00. The lowest score for girls in this group was 72.00 whereas the lowest score for boys was 87.00 (Table 3). These results indicate that the math performance of boys was better in a class of 20 or more students compared to the performance of girls in the same class size.

In summary, the above descriptive statistics indicate that the performance of boys in math becomes better as the class size increases whereas the performance of girls in math deteriorates as the class size increases.

## Factorial ANOVA

A factorial ANOVA was conducted at 95% CI and a significance level of.05 to determine the presence of the main effect and interaction between gender and classroom size on math performance. It was established that gender had a main effect on math performance since the F value for gender was significant, F (1, 54) = 19.056, p =.001 (Table 5). The F value was large enough to indicate that big differences in math performance existed across the genders. It was not necessary to perform posthoc tests for gender since gender has less than three groups which is the minimum number of groups needed for post hoc analysis.

The analysis also indicated that there was a main effect of classroom size on math performance for both boys and girls. This is indicated by a large F value which is significant, F (2, 54) = 25.311, p =.001. On conducting a post hoc test for the same, the LSD test showed significant differences (p <.05) between performances of classes of 10 or fewer students and those of 11 to 19 and 20 or more students. The Games-Howell test also confirmed significant differences in math scores between classes of 10 or fewer students and 11-19 students and 20 or more students. However, the Games-Howell test indicated non-significant differences in math performance for between classes of 11-19 students and 20 or more students (p =.086) (Table 6).

The above factorial ANOVA test also indicated an interaction between gender and classroom size and the interaction was significant, F (2, 54) = 19.102, p =.001. The R squared for this model (interaction) was.666 (Table 5) indicating that both gender and classroom size contributed to 66.6 percent of performance in math.

From the descriptive statistics and the factorial ANOVA, it is evident that math performance for children is significantly affected by the classroom size. In particular, the performance of girls is better than that of boys in classrooms with fewer students. This is supported by the fact that the mean performance for girls was better than that of boys in classrooms of 10 or fewer students and it worsened as the class size increased. On the other hand, the performance of boys bettered as the classroom size increased. The existence of the main effect of gender and classroom size as well as the interaction between the two variables also confirms this. As a result, the researcher’s hypothesis that girls would perform better than boys in classrooms with fewer students is supported.

## Applying Analytical Strategies to an Area of Research Interest

My area of research interest is to understand criminal behavior to curb crime, terrorist attacks to be specific, which may occur within or outside our borders. I would like to predict the possibility of an individual getting involved in a crime based on the individual’s background as well as personality. Childhood delinquency and parent-child relationship will be important factors to consider in determining the likelihood of engaging in criminal behaviors. It is hypothesized that a high number of childhood delinquency cases as well as poor child-parent relationships increase the likelihood of an individual getting involved in crime.

Based on my area of interest, age would be a good example of a predictor variable whereas appropriate dependent variables would be a frequency of involvement in crime, prior delinquency as well as a parent-child relationship. A Hypothetical ANCOVA indicating that the age of the individual as well as prior delinquency (covariate) significantly predicts the frequency of involvement in crime (dependent variable) was conducted. The mock output table supporting this relationship is provided below:

Table 7: Hypothetical ANCOVA Output Table for Predicting Frequency/Likelihood of being caught in Crime using Age and Prior Delinquency.

 Tests of Between-Subjects Effects Dependent Variable: Frequency of being caught in a crime Source Type III Sum of Squares df Mean Square F Sig. Corrected Model 82.563a 3 27.521 116.200 .068 Intercept 15.102 1 15.102 63.765 .079 age .000 0 . . . age * prir_del 41.763 1 41.763 176.333 .048 Error .237 1 .237 Total 198.000 5 Corrected Total 82.800 4 a. R Squared =.997 (Adjusted R Squared =.989)

Table 7 indicates that age and prior delinquency are significant predictors of an individual being involved in crime, F (1, 1) = 176.333, p =.048. The interaction of age and the covariate (prior delinquency) is very strong in determining the possibility of an individual being involved in crime as indicated by a large F value which is significant at the level of.05. The R squared value is.997 indicating that 99.7 percent of the frequency/likelihood of being caught or involved in crime is a factor of age and prior delinquency of the individual.

## Reference

Field, A P. (2009). Discovering statistics using SPSS, Third edition. San Diego, CA: Sage Publications Ltd.

## Appendix

Table 1: Descriptive Statistics for Math Performance in Classrooms of 10 or Less Students.

 Descriptivesa Gender Statistic Std. Error Math_Score Female Mean 93.8000 1.24544 95% Confidence Interval for Mean Lower Bound 90.9826 Upper Bound 96.6174 5% Trimmed Mean 93.8889 Median 94.0000 Variance 15.511 Std. Deviation 3.93841 Minimum 88.00 Maximum 98.00 Range 10.00 Interquartile Range 8.25 Skewness -.250 .687 Kurtosis -1.688 1.334 Male Mean 92.7000 1.08577 95% Confidence Interval for Mean Lower Bound 90.2438 Upper Bound 95.1562 5% Trimmed Mean 92.6667 Median 92.5000 Variance 11.789 Std. Deviation 3.43350 Minimum 87.00 Maximum 99.00 Range 12.00 Interquartile Range 4.75 Skewness .195 .687 Kurtosis .331 1.334 a. Classroom size = 10 or less

Table 2: Descriptive Statistics for Math Performance in Classrooms of 11-19 Students.

 Descriptivesa Gender Statistic Std. Error Math_Score Female Mean 88.5000 1.25831 95% Confidence Interval for Mean Lower Bound 85.6535 Upper Bound 91.3465 5% Trimmed Mean 88.5000 Median 89.0000 Variance 15.833 Std. Deviation 3.97911 Minimum 82.00 Maximum 95.00 Range 13.00 Interquartile Range 7.00 Skewness -.026 .687 Kurtosis -.670 1.334 Male Mean 89.7000 .76085 95% Confidence Interval for Mean Lower Bound 87.9788 Upper Bound 91.4212 5% Trimmed Mean 89.7222 Median 90.0000 Variance 5.789 Std. Deviation 2.40601 Minimum 86.00 Maximum 93.00 Range 7.00 Interquartile Range 5.00 Skewness -.278 .687 Kurtosis -1.269 1.334 a. Classroom size = 11-19

Table 3: Descriptive Statistics for Math Performance in Classrooms of 20 or More Students.

 Descriptivesa Gender Statistic Std. Error Math_Score Female Mean 79.2000 1.32330 95% Confidence Interval for Mean Lower Bound 76.2065 Upper Bound 82.1935 5% Trimmed Mean 79.2222 Median 80.0000 Variance 17.511 Std. Deviation 4.18463 Minimum 72.00 Maximum 86.00 Range 14.00 Interquartile Range 6.50 Skewness -.335 .687 Kurtosis -.207 1.334 Male Mean 91.2000 1.01980 95% Confidence Interval for Mean Lower Bound 88.8930 Upper Bound 93.5070 5% Trimmed Mean 91.0556 Median 90.5000 Variance 10.400 Std. Deviation 3.22490 Minimum 87.00 Maximum 98.00 Range 11.00 Interquartile Range 4.50 Skewness .918 .687 Kurtosis 1.016 1.334 a. Classroom size = 20 or more

Table 4: Between Subject Factors for Classroom Size and Gender.

 Between-Subjects Factors Value Label N Classroom size 1 10 or less 20 2 11-19 20 3 20 or more 20 Gender F Female 30 M Male 30

Table 5: Between-Subjects Effects for Classroom and Gender.

 Tests of Between-Subjects Effects Dependent Variable:Math_Score Source Type III Sum of Squares df Mean Square F Sig. Corrected Model 1381.483a 5 276.297 21.576 .000 Intercept 477220.017 1 477220.017 3.727E4 .000 Classroom 648.233 2 324.117 25.311 .000 Gender 244.017 1 244.017 19.056 .000 Classroom * Gender 489.233 2 244.617 19.102 .000 Error 691.500 54 12.806 Total 479293.000 60 Corrected Total 2072.983 59 a. R Squared =.666 (Adjusted R Squared =.636)

Table 6: Post-hoc Tests for Classroom Size.

 Multiple Comparisons Dependent Variable:Math_Score (I) Classroom size (J) Classroom size Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval Lower Bound Upper Bound LSD 10 or less 11-19 4.1500* 1.13162 .001 1.8812 6.4188 20 or more 8.0500* 1.13162 .000 5.7812 10.3188 11-19 10 or less -4.1500* 1.13162 .001 -6.4188 -1.8812 20 or more 3.9000* 1.13162 .001 1.6312 6.1688 20 or more 10 or less -8.0500* 1.13162 .000 -10.3188 -5.7812 11-19 -3.9000* 1.13162 .001 -6.1688 -1.6312 Games-Howell 10 or less 11-19 4.1500* 1.09250 .001 1.4843 6.8157 20 or more 8.0500* 1.79396 .000 3.6131 12.4869 11-19 10 or less -4.1500* 1.09250 .001 -6.8157 -1.4843 20 or more 3.9000 1.75694 .086 -.4602 8.2602 20 or more 10 or less -8.0500* 1.79396 .000 -12.4869 -3.6131 11-19 -3.9000 1.75694 .086 -8.2602 .4602 Based on observed means. The error term is Mean Square(Error) = 12.806. *. The mean difference is significant at the 0.05 level.

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