Chapter 13 - Factorial Analysis of Variance Note: Because of severe rounding in reporting and using means, there will be visible rounding error in the following answers, when compared to standard computer solutions. I have made the final answer equal the correct answer, even if that meant that it is not exactly the answer to the calculations shown. (e.g. 3(3.3) would be shown as 10.0, not 9.9) 13.1 Mother/infant interaction for primiparous/multiparous mothers under or over 18 years of age with LBW or full-term infants: Table of cell means Size/Age LBW LBW NBW < 18 > 18 Mother s Primi- 4.5 5.3 6.4 5.40 Parity Multi- 3.9 6.9 8. 6.33 4. 6.1 7.3 5.87 Σ X ( Σ X ) N ( i ) SSParity nsσ X. X.. size 404 35 338. 93 60 ( ) 10(3)[ 5.40 5.87 + 6.33 5.87 ] 30 0.4356 13.067 (. j..) SS npσ X X cells ( ) 10 [(4.00 5.87) + 6.10 5.87 + 7.30 5.87 ] 0.79 + 0.05 +.04 0 4.89 97.733 ( ij..) SS nσ X X ( ) 10[ 4.5 5.87 +... + 8. 5.87 ] 10 1.853 18.53 SS SS SS SS 18.53 13.067 97.733 PS cells P S 17.733 SSerror SStotal SScells 338.93 18.53 10.40
Source df SS MS Parity 1 13.067 13.067 3.354 Size/Age 97.733 48.867 1.541* P x S 17.733 8.867.76 Error 54 10.400 3.896 Total 59 338.933 *p <.05 F.05 (,54) 3.17 13.3 The mean for these primiparous mothers would not be expected to be a good estimate of the mean for the population of all primiparous mothers because 50% of the population of primiparous mothers do not give birth to LBW infants. This would be important if we wished to take means from this sample as somehow representing the population means for primiparous and multiparous mothers. 13.5 Memory of avoidance of a fear-producing stimulus: Area of Stimulation Neutral Area A Area B Mean 50 8.6 16.8 4.4 3.7 Delay 100 8.0 3.0 16.0.33 150 8.0 6.8 6.4 7.07 Mean 8...7 4. ΣX 1090 ΣX 8374 N 45 n 5 a 3 d 3 Σ X ( Σ X ) 8374 1090 1971. 778 N 45 ( i ) SSDelay naσ X. X.. Area ( ) 5(3)[(3.7 4.) +.33 4. + 7.07 4. ] 5 3 0.90 + 3.57 + 8.1 30 1.60 188.578 (. j..) SS ndσ X X 5(3)[(8.0 4.).0 4..7 4. ] + + 356.044 ( ij..) SS nσ X X Cells 5[(8.60 4.) 16.80 4.... 6.4 4. ] + + + 916.578 F
SS DA SS cells SS D SS A 916.578 188.578 356.044 371.956 SS error SS cells 1971.778 916.578 1055.00 Source df SS MS Delay 188.578 94.89 3. Area 356.044 178.0 6.07* D x A 4 371.956 9.989 3.17* Error 36 1055.00 9.311 Total 44 1971.778 p <. 05 [ F. 05 (, 36 ) 3. 7; F. 05 ( 4, 36 ). 64] F 13.7 In Exercise 13.5, if A refers to Area: α 1 the treatment effect for the Neutral site X.1 X.. 8. 4. 3.978 13.9 The Bonferroni test to compare Site means. t N A t N B MS error n N n A MS error n N + MS error n B 8. 0. 0 9. 311 15 + 9. 311 15 8. 0. 7 9. 311 15 3. 03 ( Reject H 0 ) 3. 00 ( Reject H 0 ) + 9. 311 15 [t.05 (,36) ±.34] We can conclude that both the difference between Groups N and A and between Groups N and B are significant, and our familywise error rate will not exceed α.05. 13.11 Rerunning Exercise 11.3 as a factorial design: The following printout is from SPSS
Dependent Variable: Recall Tests of Between-Subjects Effects Source Corrected Model Intercept Age LevelProc Age * LevelProc Error Total Corrected Total Type III Sum of Squares df Mean Square F Sig. 1059.800 a 3 353.67 53.301.000 5017.600 1 5017.600 757.056.000 115.600 1 115.600 17.44.000 79.100 1 79.100 119.51.000 15.100 1 15.100.949.000 38.600 36 6.68 6316.000 40 198.400 39 a. R Squared.816 (Adjusted R Squared.801) [ The Corrected Model is the sum of the main effects and interaction. The Intercept is the correction factor, which is (ΣX). The Total (as opposed to Corrected Total) is ΣX. The Corrected Total is what we have called Total.] Estimated Marginal Means Dependent Variable: Recall 3. Age * LevelProc 95% Confidence Interval Age 1.00.00 LevelProc 1.00.00 1.00.00 Mean Std. Error Lower Bound Upper Bound 6.500.814 4.849 8.151 19.300.814 17.649 0.951 7.000.814 5.349 8.651 1.000.814 10.349 13.651 The results show that there is a significance difference between younger and older subjects, that there is better recall in tasks which require more processing, and that there is an interaction between age and level of processing (LevelProc). The difference between the two levels of processing is greater for the younger subjects than it is for the older ones, primarily because the older ones do not do much better with greater amounts of processing.
13.13 Made-up data with main effects but no interaction: Cell means: 8 1 4 6 1 J B Row 1 10 J Row 8 B 6 J 4 B 0 Column Column Column 13.15 The interaction was of primary interest in an experiment by Nisbett in which he showed that obese people varied the amount of food they consumed depending on whether a lot or a little food was visible, while normal weight subjects ate approximately the same amount under the two conditions. 13.17 Unequal sample sizes Klemchuk, Bond, & Howell (1990): Cell ns: Daycare Age Younger Older No 14 1 6 Yes 10 4 14 4 16 40 N n h k 4 7.95 1 1 1 1 1 Σ + + + n i 14 1 10 4
Cell Means: Age Younger Older No -1.089 0.0750-0.5669 Daycare Yes -0.5631 0.5835 0.010-0.8860 0.39 13.895 (... ) *7.95 ( 0.5669 (.784)) ( 0.010 (.784)) SSDaycare anhσ X i X + *7.95*.0.1665.639 (... ) *7.95 ( 0.8860 (.784)) ( 0.39 (.784)) SScells nhσ ( X ij X.. ) 7.95 7.95*1.8146*14.3811 SS Age dnhσ X j X + *7.95*0.7384 11.704 ( 1.089 (.784)) ( 0.0750 (.784)) ( 0.5631 (.784)) + ( 0.5835 (.784) ) + + SS SS SS SS 14.381.639 11.704 0.038 AD cells D A Cell variances: Age Younger Older No 0.7407 0.1898 Daycare Yes 0.9551 0.456 ( ) SS Σ n 1 s 13(0.7407) + 11(0.1898) + 9(0.9551) + 3(0.456) 1.050 error ij ij Source df SS MS F Age 1 11.704 11.704 0.0* Daycare 1.639.639 4.51* A D 1 0.038 0.038 <1 Error 36 1.050 0.585 Total 39 * p <.05 [ F 4.11].05,(1,36) We can conclude that there are effects due to both Age and Daycare, but there is no interaction between the two main effect variables. (One variance is nearly 4
times another, but with such small sample sizes we can t tell if there is heterogeneity of variance.)? 13.19 Magnitude of effect for mother-infant interaction data in Exercise 13.1: η P SS parity 13.067 338.933.04 η S SS size 97.733 338.933.9 η PS SS PS 17.733 338.933.05 ω SS (p 1)MS parity error 13.067 (1)(3.896) P 338.933 + 3.896.03 ω SS (s 1)MS size error 97.733 ()(3.896) S 338.933 + 3.896.6 ω SS (p 1)(s 1)MS PS error 17.733 (1)()(3.896) PS 338.933 + 3.896 13.1 Magnitude of effect for avoidance learning data in Exercise 13.5: η D SS delay η A SS Area η DA SS DA 188. 578 1971. 778. 10 356. 044 1971. 778. 18 371. 956 1971. 778. 19.03 ω D SS delay ( d 1 ) MS error 188. 578 ( ) ( 9. 311) 1971. 778 + 9. 311. 06 ω A SS area ( a 1 ) MS error 356. 044 ( ) ( 9. 311) 1971. 778 + 9. 311. 15
ω DA SS DA ( d 1 ) ( a 1 ) MS error 371. 956 ( ) ( ) ( 9. 311) 1971. 778 + 9. 311. 13 13.3 Three-way ANOVA on Early Experience x Intensity of UCS x Conditioned Stimulus (Tone or Vibration): n 5 in all cells 41,151.00 E I C Cells CS Tone CS Vibration Exper Hi Med Low Hi Med Low : Contr 11 16 1 1.0 19 4 9 4.00 0.00 ol Tone 5 8 34 9.0 1 6 31 6.00 7.50 Vib 6 13 0 13.0 40 41 5 44.33 8.67 Both 30 30 7.33 35 38 48 40.33 33.83 16 1.75 105 1.33 8.75 3.5 40.00 33.66 7.50 E I Cells Intensity Experience: High Med Low Control 15 0 5 0.00 Tone 3 7 3.5 7.50 Vib 3 7 36 8.67 Both 8.5 34 39 33.83.38 7.00 33.1 7.50 E C Cells Conditioned Stimulus Experience: Tone Vib Control 16.00 4.00 0.00 Tone 9.00 6.00 7.50 Vib 13.00 44.33 8.67 Both 7.33 40.33 33.83 1.33 33.66 7.50 I C Cells Conditioned Stim Intensity: Tone Vib High 16.00 8.75.38 Med 1.75 3.5 7.00 Low 6.5 40.00 33.1 1.33 33.67 7.50
( ) 0 7.5 7.5 7.5 i 5 3 SSE nicσ X.. X... 931.667 ( 8.67 7.5) + ( 33.83 7.5) (..... ) 5( 4)( ) j (.38 7.5) ( 7.00 7.5) ( 33.1 7.5) (.... ) ij ( ) SSI necσ X X + + 36.50 + + SScellsEI ncσ X X 5 15.00 7.50 +... + ( 39.00 7.50) 535.000 SS SS SS SS 535.000 931.667 36.50 67.083 E I cellsei E I ( ) 5( 4)( 3) k ( 1.33 7.5) ( 33.66 7.5) SSC neiσ X.. X... + 4563.333 ( ) ( 5)( 3) i k ( 16.00 7.50 )... ( 40.33 7.50) SScellsEC niσ X. X... + + 1,110.000 SS SS SS SS 1,110.000 931.667 4563.333 4615.000 E C cellsec E C (....) ( 5)( 4) ij ( 15.00 7.50 )... ( 39.00 7.50) SScellsIC neσ X X + + 6945.000 SS SS SS SS 6945.000 36.50 4563.333 55.417 I C cellsic I C ijk SScellsEIC nσ X X... 5 11.00 7.50 +... + 48.00 7.50 14,680.000 SS SS SS SS SS SS SS SS E I C cellseic E I C EI EC IC 14,680.000 931.667 36.50 4563.333 67.083 4615.000 55.417 11.5
SSerror SStotal SS CellsC E I 41,151.000 14, 680.000 6.471.000 Source df SS MS F Experience 3 931.667 977. 3.544* Intensity 36.50 1163.15 4.18* Cond Stim 1 4563.333 4563.333 16.550* E x I 6 67.083 11.181 <1 E x C 3 4615.000 1538.333 5.579* I x C 55.417 7.708 <1 E x I x C 6 11.50 0.08 <1 Error 96 6,471.000 75.740 Total 119 41,151.000 p <. 05 [ F. 05 ( 1, 96 ) 3. 94; F. 05 (, 96 ) 3. 09; F. 05 ( 3, 96 ). 70; F. 05 ( 6, 96 ). 19] There are significant main effects for all variables with a significant Experience Conditioned Stimulus interaction. 13.5 Analysis of Epineq.dat: Dependent Variable: Trials to reversal Tests of Between-Subjects Effects Source Corrected Model Intercept DOSE DELAY DOSE * DELAY Error Total Corrected Total Type III Sum of Squares df Mean Square F Sig. 141.130 a 8 17.641 8.158.000 1153.787 1 1153.787 533.554.000 133.130 66.565 30.78.000.96 1.148.531.590 5.704 4 1.46.659.6 14.083 99.16 1509.000 108 355.13 107 a. R Squared.397 (Adjusted R Squared.349)
13.7 Tukey on Dosage data from Exercise 13.5 Dependent Variable: Trials to reversal Tukey HSD Multiple Comparisons (I) dosage of epinephrine 0.0 mg/kg 0.3 mg/kg 1.0 mg/kg (J) dosage of epinephrine 0.3 mg/kg 1.0 mg/kg 0.0 mg/kg 1.0 mg/kg 0.0 mg/kg 0.3 mg/kg Mean Difference (I-J) Std. Error Sig. -1.67*.35.000 1.03*.35.010 1.67*.35.000.69*.35.000-1.03*.35.010 -.69*.35.000 Based on observed means. *. The mean difference is significant at the.05 level.. All of these groups differed from each other at p <.05. 13.9 Simple effects on data in Exercise 13.8. Source df SS MS F Condition 1 918.750 918.75 34.4* Cond @ Inexp. 1 1014.00 1014.00 37.99* Cond @ Exp. 1 11.50 11.50 4.55* Cond*Exper 1 16.750 16.75 8.1* Other Effects 9 631.417 Error 36 961.000 6.694 Total 47 477.917 *P <.05 [F.05(1.36) 4.1]
13.31 Dress codes and Performance total Code (.. ) SS Σ X X School ( Yes ). (91 7.050) + (78 7.050) +... + (56 7.050) 13554.65 ( i...) SS ncσ X X ( j..) + 10* 7[(73.99 7.050) (70.171 7.050) ] 494.90 SS nσ X X + + + School ( No). 10[(79.7 73.99) (71.5 73.99)... (73.5 ( ) 10 147.414 1474.14 ( j..) SS nσ X X ( ) 73.99) ] 10[(68.5 70.171) (73.7 70.171)... (71.1 70.171) ] 10 16.314 163.14 SS SS + SS 1474.14 + 163.14 737.8 School ( Code) School ( Yes ) School ( No) SS error + + + SS SS SS ( ) 13554.65 494.9 737.8 1033.08 total C S C Source df SS MS F 1 494.90 494.90.166 1 737.80 8.107 1 737.80 88.107.784* 16 1033.08 81.931 Code Error 1 School(Code) Error Total 139 13554.65 * p <.05 The F for Code is not significant but the F for the nested effect is. But notice that the two F values are not all that far apart but their p values are very different. The reason for this is that we only have 1 df for error to test Code, but 16 df for error to test School(Code).
13.33 Analysis of Seligman et al. (1990) If we think that males are generally more optimistic than females, then the sample sizes themselves are part of the treatment effect. We probably would not want to ignore that if we are looking at sex as an independent variable. In fact, the lack of independence between sample size and the effect under study is an important problem when it occurs. 13.35 This question does