Tutorial: Two Way ANOVA

1. A researcher wants to study the effect of exercise and diet on percentage of body fat. A sample of 20 university students is randomly chosen and randomly placed in four different treatment conditions. Results of this study is shown in the table below

Diet of less than 2000 calories

Diet of more than 2000 calories

Exercise 10, 12, 14, 12, 10 12, 14, 15, 16, 14No exercise 16, 18, 20, 22, 20 22, 24, 24, 22, 24

a) State (i) the independent variable and its levels, and (ii) the dependent variable in this study. b) State the assumptions that has to be obeyed when carrying out two Way ANOVA. c) Using SPSS, test the hypothesis whether exercise and diet has a significant effect on

percentage of body fat, at a significance level of .01.

2. A researcher wants to study the effect of teaching method and gender on statistics performance of students. A sample of 36 university students was randomly chosen and randomly placed in six different treatment conditions. The statistics performance score is shown in the table below.

Male FemaleMethod I 32, 30, 36, 38, 38, 35 25, 29, 27, 26, 28, 24Method II 34, 40, 36, 31, 39, 33 40, 38, 29, 34, 33, 37Method III 40, 35, 42, 38, 36, 41 43, 36, 41, 39, 37, 45

a) State (i) the independent variable and its levels, and (ii) the dependent variable in this study. b) State the assumptions that has to be obeyed when carrying out two Way ANOVA. c) Using SPSS, test the hypothesis whether teaching method and gender has a significant effect

on statistics performance, at a significance level of .05.d) Is a Post Hoc test necessary? Explain, aand if necessary use a Tukey test.

3. A researcher wants to study the effect of reward and gender on mathematics problem solving ability (kebolehan penyelesaian masalah matematik) of students. A sample of 32 students was randomly chosen and randomly placed in four different treatment conditions. The mathematics problem solving score is shown in the table below.

Male (Lelaki) Female (Perempuan)Reward (Ganjaran)

21, 22, 19, 23, 17, 24, 20, 25 6, 4, 7, 11, 5, 8, 12, 9

No reward 15, 9, 11, 19, 16, 14, 13, 12 8, 7, 3, 11, 5, 10, 5, 6

The results of this study, at .01 significance level is shown in the following tables, obtained from the SPSS output

Analysis of Variance

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Between-Subjects Factors

Ganjaran 16TiadaGanjaran 16

Lelaki 16Perempuan 16

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Ganjaran(Faktor A)

12

Jantina (FaktorB)

Value Label N

Descriptive Statistics

Dependent Variable: Kebolehan Penyelesaian Masalah Matematik

21.38 2.669 87.75 2.816 8

14.56 7.519 1613.63 3.114 86.88 2.696 8

10.25 4.480 1617.50 4.885 167.31 2.701 16

12.41 6.470 32

Jantina (Faktor B)LelakiPerempuanTotalLelakiPerempuanTotalLelakiPerempuanTotal

Ganjaran (Faktor A)Ganjaran

Tiada Ganjaran

Total

Mean Std. Deviation N

Levene's Test of Equality of Error Variancesa

Dependent Variable: Kebolehan PenyelesaianMasalah Matematik

.048 3 28 .986F df1 df2 Sig.

Tests the null hypothesis that the error variance of thedependent variable is equal across groups.

Design: Intercept+Ganjaran+Jantina+Ganjaran* Jantina

a.

Tests of Between-Subjects Effects

Dependent Variable: Kebolehan Penyelesaian Masalah Matematik

1073.594a 3 357.865 44.708 .000 .8274925.281 1 4925.281 615.317 .000 .956148.781 1 148.781 18.587 .000 .399830.281 1 830.281 103.727 .000 .78794.531 1 94.531 11.810 .002 .297

224.125 28 8.0046223.000 321297.719 31

SourceCorrected ModelInterceptGanjaranJantinaGanjaran * JantinaErrorTotalCorrected Total

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

R Squared = .827 (Adjusted R Squared = .809)a.

Profile Plots

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Ganjaran Tiada Ganjaran

Ganjaran (Faktor A)

10

15

20

Estim

ated

Mar

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l Mea

ns

Jantina (Faktor B)LelakiPerempuan

Estimated Marginal Means of Kebolehan Penyelesaian Masalah Matematik

a) State (i) the independent variable and its levels, and (ii) the dependent variable in this study. b) State the assumptions that has to be obeyed when carrying out two Way ANOVA.c) Do the scores of the dependent variable in each cell have equal variances? Explain.d) State the null hypotheses in this study. e) Is there a significant main effect of factor A on mathematics problem solving ability? Explain.f) Is there a significant main effect of factor B on mathematics problem solving ability? Explain.g) Is there a significant interaction effect between factor A and factor B on mathematics problem

solving ability? Explain. h) What is the effect size for the main effect and interaction effect that is significant? Explain.a) Should a Post Hoc be done? Explain why and if necessary, use a Tukey test.

4. A researcher wants to study the effect of teaching method (kaedah pengajaran) and IQ on students’ reading ability. A sampel of 36 students was chosen randomly and placed randomly in six different treatment situations. The reading ability score is given in the following table:

Low (Rendah) IQ Moderate (Sederhana) IQ

High (Tinggi) IQ

Method I 22, 18, 16, 20 24, 21, 20, 20 40, 36, 38, 36Method II 20, 18, 19, 15 20, 29, 25, 30 32, 29, 36, 35

The results of this study, at .05 significance level is shown in the following tables, obtained from the SPSS output

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Univariate Analysis of Variance

Between-Subjects Factors

Kaedah I 12Kaedah II 12Rendah 8Sederhana 8

Tinggi 8

12

Kaedah Pengajaran(Faktor A)

123

IQ (Faktor B)

Value Label N

Descriptive Statistics

Dependent Variable: Kebolehan Membaca

19.00 2.582 421.25 1.893 437.50 1.915 425.92 8.826 1218.00 2.160 426.00 4.546 433.00 3.162 425.67 7.114 1218.50 2.268 823.62 4.104 835.25 3.412 825.79 7.841 24

IQ (Faktor B)RendahSederhanaTinggiTotalRendahSederhanaTinggiTotalRendahSederhanaTinggiTotal

Kaedah Pengajaran(Faktor A)Kaedah I

Kaedah II

Total

Mean Std. Deviation N

Levene's Test of Equality of Error Variancesa

Dependent Variable: Kebolehan Membaca

1.506 5 18 .237F df1 df2 Sig.

Tests the null hypothesis that the error variance ofthe dependent variable is equal across groups.

Design:Intercept+KaedahPengajaran+IQ+KaedahPengajaran * IQ

a.

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Tests of Between-Subjects Effects

Dependent Variable: Kebolehan Membaca

1266.208a 5 253.242 30.852 .000 .89615965.042 1 15965.042 1944.980 .000 .991

.375 1 .375 .046 .833 .0031178.583 2 589.292 71.792 .000 .889

87.250 2 43.625 5.315 .015 .371147.750 18 8.208

17379.000 241413.958 23

SourceCorrected ModelInterceptKaedahPengajaranIQKaedahPengajaran * IQErrorTotalCorrected Total

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

R Squared = .896 (Adjusted R Squared = .866)a.

Post Hoc TestsIQ (Faktor B)

Multiple Comparisons

Dependent Variable: Kebolehan MembacaTukey HSD

-5.12* 1.433 .006 -8.78 -1.47-16.75* 1.433 .000 -20.41 -13.09

5.12* 1.433 .006 1.47 8.78-11.63* 1.433 .000 -15.28 -7.9716.75* 1.433 .000 13.09 20.4111.63* 1.433 .000 7.97 15.28

(J) IQ (Faktor B)SederhanaTinggiRendahTinggiRendahSederhana

(I) IQ (Faktor B)Rendah

Sederhana

Tinggi

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound95% Confidence Interval

Based on observed means.The mean difference is significant at the .05 level.*.

Homogeneous Subsets

Kebolehan Membaca

Tukey HSDa,b

8 18.508 23.628 35.25

1.000 1.000 1.000

IQ (Faktor B)RendahSederhanaTinggiSig.

N 1 2 3Subset

Means for groups in homogeneous subsets are displayed.Based on Type III Sum of SquaresThe error term is Mean Square(Error) = 8.208.

Uses Harmonic Mean Sample Size = 8.000.a.

Alpha = .05.b.

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Profile Plots

Kaedah I Kaedah II

Kaedah Pengajaran (Faktor A)

20

25

30

35

Estim

ated

Mar

gina

l Mea

ns

IQ (Faktor B)RendahSederhanaTinggi

Estimated Marginal Means of Kebolehan Membaca

a) State (i) the independent variable and its levels, and (ii) the dependent variable in this study. b) State the assumptions that has to be obeyed when carrying out two Way ANOVA.c) Do the scores of the dependent variable in each cell have equal variances? Explain.d) State the null hypotheses in this study. e) Is there a significant main effect of factor A on reading ability? Explain.f) Is there a significant main effect of factor B on reading ability? Explain.g) Is there a significant interaction effect between factor A and factor B on reading ability?

Explain. h) What is the effect size for the main effect and interaction effect that is significant? Explain.i) Should a Post Hoc be done? Explain why and if necessary, use a Tukey test.

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