One Way ANOVA with Tukey Post hoc. Case Processing Summary

Transcription

1 EXAMINE VARIABLES=Score BY Group /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Group One Way ANOVA with Tukey Post hoc Case Processing Summary Score Cases Valid Missing Total Group N Percent N Percent N Percent % 0.0% % % 0.0% % % 0.0% % % 0.0% % Descriptives Score Group Statistic Std. Error 1 Mean % Confidence Lower Bound 2.83 Interval for Mean Upper Bound % Trimmed Mean 6.00 Median 6.00 Variance Std. Deviation Minimum 3 Maximum 9 Range 6 Interquartile Range 5 Skewness Kurtosis Mean % Confidence Lower Bound 7.48 Interval for Mean Upper Bound % Trimmed Mean 8.94 Median 9.00

2 3 4 Variance Std. Deviation Minimum 8 Maximum 11 Range 3 Interquartile Range 2 Skewness Kurtosis Mean % Confidence Lower Bound 5.04 Interval for Mean Upper Bound % Trimmed Mean 7.00 Median 7.00 Variance Std. Deviation Minimum 5 Maximum 9 Range 4 Interquartile Range 3 Skewness Kurtosis Mean % Confidence Lower Bound.32 Interval for Mean Upper Bound % Trimmed Mean 2.33 Median 2.00 Variance Std. Deviation Minimum 1 Maximum 5 Range 4 Interquartile Range 3 Skewness Kurtosis Score Kolmogorov-Smirnov(a) Tests of Normality Shapiro-Wilk Group Statistic df Sig. Statistic df Sig (*) (*) (*) * This is a lower bound of the true significance. a Lilliefors Significance Correction

3 Test the Shapiro-Wilk statistic Score UNIANOVA Score BY Group /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /POSTHOC = Group ( TUKEY ) /PRINT = DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA = ALPHA(.05) /DESIGN = Group.

4 Univariate Analysis of Variance Between-Subjects Factors Group N Dependent Variable: Score Descriptive Statistics Group Mean Std. Deviation N Total Levene's Test of Equality of Error Variances(a) Dependent Variable: Score F df1 df2 Sig Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a Design: Intercept+Group HOV test at.01. Dependent Variable: Score Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Corrected Model (a) Intercept Group Error Total Corrected Total a R Squared =.683 (Adjusted R Squared =.624) Use the row with the name of the IV to interpret the ANOVA results. A statistically significant difference amongthe means is evident.

5 Post Hoc Tests Group Multiple Comparisons Dependent Variable: Score Tukey HSD (I) Group Mean 95% Confidence Interval Difference (J) Group (I-J) Std. Error Sig. Lower Bound Upper Bound (*) (*) (*) (*) (*) (*) Based on observed means. * The mean difference is significant at the.05 level. Tukey post hoc analysis to determine where the differences exist between the groups. Homogeneous Subsets Tukey HSD Score Subset Group N Sig Means for groups in homogeneous subsets are displayed. Based on Type III Sum of Squares The error term is Mean Square(Error) = a Uses Harmonic Mean Sample Size = b Alpha =.05.

6 A priori ANOVA EXAMINE VARIABLES=Score BY Group /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Group Case Processing Summary Score Cases Valid Missing Total Group N Percent N Percent N Percent % 0.0% % % 0.0% % % 0.0% % % 0.0% % Descriptives Score Group Statistic Std. Error 1 Mean % Confidence Lower Bound 2.83 Interval for Mean Upper Bound % Trimmed Mean 6.00 Median 6.00 Variance Std. Deviation Minimum 3 Maximum 9 Range 6 Interquartile Range 5 Skewness Kurtosis Mean % Confidence Lower Bound 7.48 Interval for Mean Upper Bound % Trimmed Mean 8.94 Median 9.00

7 3 4 Variance Std. Deviation Minimum 8 Maximum 11 Range 3 Interquartile Range 2 Skewness Kurtosis Mean % Confidence Lower Bound 5.04 Interval for Mean Upper Bound % Trimmed Mean 7.00 Median 7.00 Variance Std. Deviation Minimum 5 Maximum 9 Range 4 Interquartile Range 3 Skewness Kurtosis Mean % Confidence Lower Bound.32 Interval for Mean Upper Bound % Trimmed Mean 2.33 Median 2.00 Variance Std. Deviation Minimum 1 Maximum 5 Range 4 Interquartile Range 3 Skewness Kurtosis Score Kolmogorov-Smirnov(a) Tests of Normality Shapiro-Wilk Group Statistic df Sig. Statistic df Sig (*) (*) (*) * This is a lower bound of the true significance. a Lilliefors Significance Correction

8 Score UNIANOVA Score BY Group /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /POSTHOC = Group ( TUKEY ) /PRINT = DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA = ALPHA(.05) /DESIGN = Group.

9 Univariate Analysis of Variance Between-Subjects Factors Group N Dependent Variable: Score Descriptive Statistics Group Mean Std. Deviation N Total Levene's Test of Equality of Error Variances(a) Dependent Variable: Score F df1 df2 Sig Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a Design: Intercept+Group Dependent Variable: Score Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Corrected Model (a) Intercept Group Error Total Corrected Total a R Squared =.683 (Adjusted R Squared =.624) ONEWAY Score BY Group /CONTRAST= /CONTRAST= /MISSING ANALYSIS.

10 Contrast Coefficients Group Contrast Contrast Tests Score Assume equal variances Contrast Value of Contrast Std. Error t df Sig. (2-tailed) Does not assume equal variances The t-tests serve as the a priori analyses after a significant ANOVA. Because the HOV test was not significant, assume equal variances.

11 Factorial ANOVA with No Interaction EXAMINE VARIABLES=gpaimpr BY gender method /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Notes Output Created 20-DEC :17:55 Comments Input Missing Value Handling Syntax Data C:\Documents and Settings\BalkinRick\Desktop\613\SPS S data\windows\lesson 25\Lesson 25 Data File 1.sav Active Dataset DataSet4 Filter <none> Weight <none> Split File <none> N of Rows in Working Data File 60 Definition of Missing Cases Used User-defined missing values for dependent variables are treated as missing. Statistics are based on cases with no missing values for any dependent variable or factor used. EXAMINE VARIABLES=gpaimpr BY gender method /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Resources Elapsed Time 0:00:03.76

12 Gender Case Processing Summary Change in GPA Cases Valid Missing Total Gender N Percent N Percent N Percent Men % 0.0% % Women % 0.0% % Descriptives Change in GPA Gender Statistic Std. Error Men Mean % Confidence Lower Bound.2792 Interval for Mean Upper Bound.4808 Women 5% Trimmed Mean.3741 Median.3500 Variance.073 Std. Deviation Minimum -.10 Maximum 1.00 Range 1.10 Interquartile Range.41 Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound.1228 Upper Bound % Trimmed Mean.1870 Median.2000 Variance.036 Std. Deviation Minimum -.10 Maximum.60 Range.70 Interquartile Range.33 Skewness Kurtosis

13 Tests of Normality Change in GPA Kolmogorov-Smirnov(a) Shapiro-Wilk Gender Statistic df Sig. Statistic df Sig. Men (*) Women * This is a lower bound of the true significance. a Lilliefors Significance Correction Tests for normality of GPA change across gender at.01. Change in GPA

14 Note-Taking methods Case Processing Summary Change in GPA Cases Valid Missing Total Note-Taking methods N Percent N Percent N Percent Method % 0.0% % Method % 0.0% % Control % 0.0% % Descriptives Change in GPA Note-Taking methods Statistic Std. Error Method 1 Mean % Confidence Lower Bound.1502 Interval for Mean Upper Bound.3548 Method 2 Control 5% Trimmed Mean.2361 Median.2250 Variance.048 Std. Deviation Minimum.00 Maximum.80 Range.80 Interquartile Range.38 Skewness Kurtosis Mean % Confidence Lower Bound.3560 Interval for Mean Upper Bound % Trimmed Mean.4694 Median.5000 Variance.062 Std. Deviation Minimum.00 Maximum 1.00 Range 1.00 Interquartile Range.34 Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound.0662 Upper Bound % Trimmed Mean.1333

15 Median.1000 Variance.022 Std. Deviation Minimum -.10 Maximum.40 Range.50 Interquartile Range.24 Skewness Kurtosis Change in GPA Tests of Normality Kolmogorov-Smirnov(a) Shapiro-Wilk Note-Taking methods Statistic df Sig. Statistic df Sig. Method (*) Method (*) Control (*) * This is a lower bound of the true significance. a Lilliefors Significance Correction Tests for normality of GPA change across method at.01.

16 Change in GPA Boxplots UNIANOVA gpaimpr BY gender method /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /POSTHOC = method ( TUKEY ) /PLOT = PROFILE( method*gender ) /PRINT = DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA = ALPHA(.05) /DESIGN = gender method gender*method.

17 Univariate Analysis of Variance Between-Subjects Factors Gender Note-Taking methods Value Label N 1 Men 30 2 Women 30 1 Method Method Control 20 Descriptive Statistics Dependent Variable: Change in GPA Gender Note-Taking methods Mean Std. Deviation N Men Method Method Control Total Women Method Method Control Total Total Method Method Control Total Levene's Test of Equality of Error Variances(a) Dependent Variable: Change in GPA F df1 df2 Sig Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a Design: Intercept+gender+method+gender * method HOV test at.01

18 Tests of Between-Subjects Effects Dependent Variable: Change in GPA Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Corrected Model 1.889(a) Intercept gender method gender * method Error Total Corrected Total a R Squared =.515 (Adjusted R Squared =.470) First, look at the interaction effect--it is not signficant. So, main effects can be interpreted. The main effects are denoted by the rows with the names of the IVs. Each of the main effects is statistically significant. Because Method has more than two levels, a Tukey post hoc is needed to determine where the differences lie. Post Hoc Tests Note-Taking methods Dependent Variable: Change in GPA Tukey HSD Multiple Comparisons (I) Note-Taking methods Method 1 Method 2 Control Based on observed means. * The mean difference is significant at the.05 level. Mean 95% Confidence Interval Difference (J) Note-Taking methods (I-J) Std. Error Sig. Lower Bound Upper Bound Method (*) Control Method (*) Control.3375(*) Method Method (*)

19 Homogeneous Subsets Tukey HSD Change in GPA Subset Note-Taking methods N 1 2 Control Method Method Sig Means for groups in homogeneous subsets are displayed. Based on Type III Sum of Squares The error term is Mean Square(Error) =.033. a Uses Harmonic Mean Sample Size = b Alpha =.05.

20 Profile Plots I have included this plot for instructional purposes, as it is not necessary since the interaction effect was not significant. Note the similarity in the patterns of responses.

21 Factorial ANOVA with Significant Interaction EXAMINE VARIABLES=gpaimpr BY gender method /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Gender Case Processing Summary Change in GPA Cases Valid Missing Total Gender N Percent N Percent N Percent Men % 0.0% % Women % 0.0% % Descriptives Change in GPA Gender Statistic Std. Error Men Mean % Confidence Lower Bound.1934 Interval for Mean Upper Bound.3433 Women 5% Trimmed Mean.2611 Median.2500 Variance.040 Std. Deviation Minimum -.10 Maximum.80 Range.90 Interquartile Range.30 Skewness Kurtosis Mean % Confidence Lower Bound.1958 Interval for Mean Upper Bound % Trimmed Mean.2907 Median.2250 Variance.086 Std. Deviation.29254

22 Minimum -.10 Maximum 1.00 Range 1.10 Interquartile Range.53 Skewness Kurtosis Tests of Normality Kolmogorov-Smirnov(a) Shapiro-Wilk Gender Statistic df Sig. Statistic df Sig. Change in GPA Men Women a Lilliefors Significance Correction

23 Change in GPA Note-Taking methods Case Processing Summary Change in GPA Cases Valid Missing Total Note-Taking methods N Percent N Percent N Percent Method % 0.0% % Method % 0.0% % Control % 0.0% %

24 Descriptives Change in GPA Note-Taking methods Statistic Std. Error Method 1 Mean % Confidence Lower Bound.1502 Interval for Mean Upper Bound.3548 Method 2 Control 5% Trimmed Mean.2361 Median.2250 Variance.048 Std. Deviation Minimum.00 Maximum.80 Range.80 Interquartile Range.38 Skewness Kurtosis Mean % Confidence Lower Bound.3560 Interval for Mean Upper Bound % Trimmed Mean.4694 Median.5000 Variance.062 Std. Deviation Minimum.00 Maximum 1.00 Range 1.00 Interquartile Range.34 Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound.0662 Upper Bound % Trimmed Mean.1333 Median.1000 Variance.022 Std. Deviation Minimum -.10 Maximum.40 Range.50 Interquartile Range.24 Skewness Kurtosis

25 Change in GPA Tests of Normality Kolmogorov-Smirnov(a) Shapiro-Wilk Note-Taking methods Statistic df Sig. Statistic df Sig. Method (*) Method (*) Control (*) * This is a lower bound of the true significance. a Lilliefors Significance Correction Change in GPA Boxplots

26 UNIANOVA gpaimpr BY gender method /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /PLOT = PROFILE( method*gender ) /PRINT = DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA = ALPHA(.05) /DESIGN = gender method gender*method. Univariate Analysis of Variance Between-Subjects Factors Gender Note-Taking methods Value Label N 1 Men 30 2 Women 30 1 Method Method Control 20 Descriptive Statistics Dependent Variable: Change in GPA Gender Note-Taking methods Mean Std. Deviation N Men Method Method Control Total Women Method Method Control Total Total Method Method Control Total Levene's Test of Equality of Error Variances(a) Dependent Variable: Change in GPA F df1 df2 Sig Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a Design: Intercept+gender+method+gender * method

27 Tests of Between-Subjects Effects Dependent Variable: Change in GPA Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Corrected Model 1.889(a) Intercept gender method gender * method Error Total Corrected Total a R Squared =.515 (Adjusted R Squared =.470) In this example, there is a significant interaction, so main effects cannot be interpreted. Simple effects will need to be analyzed.

28 Profile Plots The interaction effect is plotted to demonstrate the discrepancy among the groups. UNIANOVA gpaimpr BY gender method /emmeans=table(gender*method) comp(method). SPSS code for analyzing simple effects.

29 Estimated Marginal Means Gender * Note-Taking methods Estimates Dependent Variable: Change in GPA 95% Confidence Interval Gender Note-Taking methods Mean Std. Error Lower Bound Upper Bound Men Method Method Control Women Method Method Control Dependent Variable: Change in GPA Pairwise Comparisons 95% Confidence Interval for Mean Difference(a) Gender (I) Note-Taking methods (J) Note-Taking methods Difference (I-J) Std. Error Sig.(a) Lower Bound Upper Bound Men Method 1 Method Control.170(*) Women Method 2 Control Method 1 Method 2 Control Method Control Method (*) Method Method (*) Control Method 1.470(*) Control.535(*) Method Method (*) Based on estimated marginal means * The mean difference is significant at the.050 level. a Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments). SPSS runs the post hoc tests before the tests for simple effects. Analyze simple effects first. Note that the simple effects (below)show statistical significance for women, but not men. Thus, we only need to look at post hoc tests for women.

30 Univariate Tests Dependent Variable: Change in GPA Gender Men Women Sum of Squares df Mean Square F Sig. Contrast Error Contrast Error Each F tests the simple effects of Note-Taking methods within each level combination of the other effects shown. These tests are based on the linearly independent pairwise comparisons among the estimated marginal means.

31 EXAMINE VARIABLES=time1 time2 time3 time4 /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Case Processing Summary Repeated Measures ANOVA Cases Valid Missing Total N Percent N Percent N Percent time % 0.0% % time % 0.0% % time % 0.0% % time % 0.0% % Descriptives time1 time2 Statistic Std. Error Mean % Confidence Lower Bound Interval for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum 46 Maximum 81 Range 35 Interquartile Range 12 Skewness Kurtosis Mean % Confidence Lower Bound Interval for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum 41

32 time3 time4 Maximum 85 Range 44 Interquartile Range 12 Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum 42 Maximum 84 Range 42 Interquartile Range 9 Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum 35 Maximum 91 Range 56 Interquartile Range 15 Skewness Kurtosis Kolmogorov-Smirnov(a) Tests of Normality Shapiro-Wilk Statistic df Sig. Statistic df Sig. time (*) time time time * This is a lower bound of the true significance. a Lilliefors Significance Correction Note that normality tests are run without a factor.

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36 GLM time1 time2 time3 time4 /WSFACTOR = time 4 Polynomial /METHOD = SSTYPE(3) /PRINT = DESCRIPTIVE ETASQ /CRITERIA = ALPHA(.05) /WSDESIGN = time.

37 General Linear Model Within-Subjects Factors Measure: MEASURE_1 Dependent time Variable 1 time1 2 time2 3 time3 4 time4 Descriptive Statistics Mean Std. Deviation N time time time time Multivariate Tests(b) Partial Eta Effect Value F Hypothesis df Error df Sig. Squared time Pillai's Trace (a) Wilks' Lambda (a) Hotelling's Trace (a) Roy's Largest Root (a) a Exact statistic b Design: Intercept Within Subjects Design: time Measure: MEASURE_1 Mauchly's Test of Sphericity(b) Epsilon(a) Within Subjects Effect Mauchly's W Approx. Chi- Square df Sig. Greenhouse- Geisser Huynh-Feldt Lower-bound time Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. b Design: Intercept Within Subjects Design: time The sphericity assumption is not met. Additionally, E <.70. So, ANOVA results should be analyzed using Greenhouse-Geisser analysis below.

38 Measure: MEASURE_1 Source time Error(time) Tests of Within-Subjects Effects Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound Since the Greenhouse-Geisser results are significant, post hoc analysis is necessary. A Tukey post hoc can be conducted by dependentt-tests. Measure: MEASURE_1 Source time Error(time) Tests of Within-Subjects Contrasts time Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Linear Quadratic Cubic Linear Quadratic Cubic Measure: MEASURE_1 Transformed Variable: Average Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Intercept Error T-TEST PAIRS = time1 time1 time1 time2 time2 time3 WITH time2 time3 time4 time3 time4 time4 (PAIRED) /CRITERIA = CI(.992) /MISSING = ANALYSIS.

39 T-Test Paired Samples Statistics Pair 1 Pair 2 Pair 3 Pair 4 Pair 5 Pair 6 Mean N Std. Deviation Std. Error Mean time time time time time time time time time time time time Paired Samples Correlations N Correlation Sig. Pair 1 time1 & time Pair 2 time1 & time Pair 3 time1 & time Pair 4 time2 & time Pair 5 time2 & time Pair 6 time3 & time Paired Samples Test Mean Paired Differences 99.2% Confidence Interval of the Difference Std. Error Std. Deviation Mean Lower Upper t df Sig. (2-tailed) Pair 1 time1 - time Pair 2 time1 - time Pair 3 time1 - time Pair 4 time2 - time Pair 5 time2 - time Pair 6 time3 - time Since repeated t-tests are conducted, a Bonferroni adjustment is needed..05 / 6 =.008. So, tests are conducted at an alpha level of.008.

40 SPANOVA GET FILE='/Users/richardbalkin/Desktop/CNEP 6372/SPANOVA data.sav'. EXAMINE VARIABLES=Pretest Posttest BY Exercisetype /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Input Missing Value Handling Notes Output Created 14-Jan :10:07 Comments Data Active Dataset Filter Weight Split File N of Rows in Working Data File Definition of Missing Cases Used /Users/richardbalkin/Desktop/CNEP 6372/SPANOVA data.sav DataSet1 <none> <none> <none> User-defined missing values for dependent variables are treated as missing. 50 Statistics are based on cases with no missing values for any dependent variable or factor used.

41 Syntax EXAMINE VARIABLES=Pretest Posttest BY Exercisetype /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Resources Processor Time 0:00: Elapsed Time 0:00: [DataSet1] /Users/richardbalkin/Desktop/CNEP 6372/SPANOVA data.sav Exercise type Case Processing Summary Cases Exercis Valid Missing Total e type N Percent N Percent N Percent Pretest Posttest % 0.0% % % 0.0% % % 0.0% % % 0.0% %

42 Pretest Descriptives Exercise type Statistic Std. Error 1 2 Posttest 1 95% Confidence Interval for Mean 95% Confidence Interval for Mean Mean Lower Bound 2.20 Upper Bound % Trimmed Mean 2.35 Median 2.45 Variance.112 Std. Deviation.335 Minimum 2 Maximum 3 Range 1 Interquartile Range 0 Skewness Kurtosis Mean Lower Bound 2.28 Upper Bound % Trimmed Mean 2.40 Median 2.40 Variance.069 Std. Deviation.264 Minimum 2 Maximum 3 Range 1 Interquartile Range 0 Skewness Kurtosis Mean % Confidence Interval Lower Bound 2.71 for Mean Upper Bound 2.80

43 2 5% Trimmed Mean 2.76 Median 2.76 Variance.011 Std. Deviation.105 Minimum 2 Maximum 3 Range 0 Interquartile Range 0 Skewness Kurtosis Mean % Confidence Interval Lower Bound 2.91 for Mean Upper Bound % Trimmed Mean 2.97 Median 2.96 Variance.017 Std. Deviation.129 Minimum 3 Maximum 3 Range 0 Interquartile Range 0 Skewness Kurtosis Tests of Normality Exercis Kolmogorov-Smirnov a Shapiro-Wilk e type Statistic df Sig. Statistic df Sig. Pretest Posttest * *

44 a. Lilliefors Significance Correction *. This is a lower bound of the true significance. Pretest Normal Q-Q Plots

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46 Detrended Normal Q-Q Plots

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49 Posttest Normal Q-Q Plots

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51 Detrended Normal Q-Q Plots

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54 GLM Pretest Posttest BY Exercisetype /WSFACTOR=time 2 Polynomial /METHOD=SSTYPE(3) /PLOT=PROFILE(time*Exercisetype) /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA=ALPHA(.05) /WSDESIGN=time /DESIGN=Exercisetype.

55 General Linear Model Input Missing Value Handling Notes Output Created 14-Jan :11:49 Comments Data Active Dataset Filter Weight Split File N of Rows in Working Data File Definition of Missing Cases Used Syntax /Users/richardbalkin/Desktop/CNEP 6372/SPANOVA data.sav DataSet1 <none> <none> <none> User-defined missing values are treated as missing. Statistics are based on all cases with valid data for all variables in the model. GLM Pretest Posttest BY Exercisetype /WSFACTOR=time 2 Polynomial /METHOD=SSTYPE(3) 50 /PLOT=PROFILE(time*Exercisetype) /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA=ALPHA(.05) /WSDESIGN=time /DESIGN=Exercisetype. Resources Processor Time 0:00: Elapsed Time 0:00: [DataSet1] /Users/richardbalkin/Desktop/CNEP 6372/SPANOVA data.sav

56 Within-Subjects Factors Measure:MEASURE_1 time 1 Pretest 2 Posttest Dependent Variable Between-Subjects Factors N Exercise type Pretest Posttest Descriptive Statistics Exercis e type Mean Std. Deviation N Total Total

57 Box's Test of Equality of Covariance Matrices a Box's M F df1 3 df Sig..332 Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. a. Design: Intercept + Exercisetype Within Subjects Design: time Multivariate Tests b Effect Value F Hypothesis df Error df Sig. time time * Exercisetype a. Exact statistic Partial Eta Squared Pillai's Trace a Wilks' Lambda a Hotelling's Trace a Roy's Largest Root a Pillai's Trace a Wilks' Lambda a Hotelling's Trace a Roy's Largest Root b. Design: Intercept + Exercisetype Within Subjects Design: time a

58 Measure:MEASURE_1 Within Subjec ts Effect Mauchly's W a Mauchly's Test of Sphericity b Approx. Chi- Square df Sig. Epsilon a Greenhouse- Geisser Huynh-Feldt Lower-bound time Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. b. Design: Intercept + Exercisetype Within Subjects Design: time

59 Measure:MEASURE_1 Source time time * Exercisetype Error(time) Sphericity Assumed Greenhouse- Geisser Tests of Within-Subjects Effects Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Huynh-Feldt Lower-bound Sphericity Assumed Greenhouse- Geisser Huynh-Feldt Lower-bound Sphericity Assumed Greenhouse- Geisser Huynh-Feldt Lower-bound Measure:MEASURE_1 Source time Tests of Within-Subjects Contrasts Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared time Linear time * Exercisetype Linear Error(time) Linear

60 Levene's Test of Equality of Error Variances a F df1 df2 Sig. Pretest Posttest Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept + Exercisetype Within Subjects Design: time Measure:MEASURE_1 Transformed Variable:Average Source Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Intercept Exercisetype Error Profile Plots

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62 EXAMINE VARIABLES=statexam /PLOT BOXPLOT HISTOGRAM NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Multiple Regression Case Processing Summary Average percentage correct on statistics exams Cases Valid Missing Total N Percent N Percent N Percent % 0.0% % Descriptives Average percentage correct on statistics exams Statistic Std. Error Mean % Confidence Lower Bound Interval for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum 23 Maximum 97 Range 74 Interquartile Range 33 Skewness Kurtosis Average percentage correct on statistics exams a Lilliefors Significance Correction Tests of Normality Kolmogorov-Smirnov(a) Shapiro-Wilk Statistic df Sig. Statistic df Sig

63 Criterion variable should be normally distributed Average percentage correct on statistics exams REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA COLLIN TOL ZPP /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT statexam /METHOD=ENTER mathtest engtest /SCATTERPLOT=(*ZRESID,*ZPRED ) /SAVE ZRESID.

64 on Descriptive Statistics Mean Std. Deviation N entage tistics test score de test Correlations elation Average percentage correct on statistics exams Average percentage correct on statistics exams Math aptitude test score English aptitude test score Math aptitude test score English aptitude test score Average percentage correct on statistics exams Math aptitude test score English aptitude test score Average percentage correct on statistics exams Math aptitude test score English aptitude test score r correlations and intercorrelations. When you divide the correlation of a predictor to the criterion by R you get the structure or that predictor.

65 ered/removed(b) riables ntered ish ude test e, Math ude test e(a) Variables Removed. Enter Method d variables entered. Variable: Average percentage correct on statistics exams Model Summary(b) R R Square Adjusted R Square Std. Error of the Estimate 505(a) Constant), English aptitude test score, Math aptitude test score Variable: Average percentage correct on statistics exams ariance accounted for in the model ANOVA(b) Sum of Squares df Mean Square F Sig. ression (a) dual l Constant), English aptitude test score, Math aptitude test score Variable: Average percentage correct on statistics exams of the model Coefficients(a) Unstandardized Coefficients Standardized Coefficients Correlations Collinear B Std. Error Beta t Sig. Zero-order Partial Part Tolerance nstant) h aptitude test score ish aptitude test e Variable: Average percentage correct on statistics exams rdized beta coefficients are the amount of increase in the criterion for each change in the predict s show statistical significance of the beta weights. quare the part correlation you have the squared semi-partial correlation coefficient-the unique amo ontributed by a predictor variable. d tolerance show no multicollinearlity noted.

66 Collinearity Diagnostics(a) ension Eigenvalue Condition Index (Constant) Variance Proportions Math aptitude test score English aptitude test score Variable: Average percentage correct on statistics exams Residuals Statistics(a) Minimum Maximum Mean Std. Deviation N ue Value Variable: Average percentage correct on statistics exams

67 have a constant variance. p between the criterion variable and each predictor variable should be linear. S=ZRE_1 XPLOT NPPLOT GROUP ICS DESCRIPTIVES AL 95 LISTWISE.

68 Case Processing Summary Cases Valid Missing Total N Percent N Percent N Percent Residual % 0.0% % Descriptives Residual Statistic Std. Error Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance.980 Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Tests of Normality Kolmogorov-Smirnov(a) Shapiro-Wilk Statistic df Sig. Statistic df Sig. Residual nificance Correction diction should be normally distributed.

69 Standardized Residual

70 MANOVA with Discriminant Analysis Post hoc GET FILE='/Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 28/Lesson 28 Data File 1.sav'. EXAMINE VARIABLES=applicat recall BY group /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Input Missing Value Handling Notes Output Created 18-Feb :42:17 Comments Data Active Dataset Filter Weight Split File N of Rows in Working Data File Definition of Missing Cases Used /Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 28/Lesson 28 Data File 1.sav DataSet1 <none> <none> <none> User-defined missing values for dependent variables are treated as missing. 30 Statistics are based on cases with no missing values for any dependent variable or factor used.

71 Syntax EXAMINE VARIABLES=applicat recall BY group /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Resources Processor Time 0:00: Elapsed Time 0:00: [DataSet1] /Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 28/Lesson 28 Data File 1.sav Study Strategy Groups Case Processing Summary Study Cases Strategy Valid Missing Total Groups N Percent N Percent N Percent Application Exam Recall Exam Think % 0.0% % Write % 0.0% % Talk % 0.0% % Think % 0.0% % Write % 0.0% % Talk % 0.0% % Descriptives Study Strategy Groups Statistic Std. Error

72 Application Exam Think Write Talk 95% Confidence Interval for Mean 95% Confidence Interval for Mean 95% Confidence Interval for Mean Mean Lower Bound 2.32 Upper Bound % Trimmed Mean 3.22 Median 3.00 Variance Std. Deviation Minimum 1 Maximum 5 Range 4 Interquartile Range 2 Skewness Kurtosis Mean Lower Bound 3.74 Upper Bound % Trimmed Mean 5.06 Median 5.00 Variance Std. Deviation Minimum 2 Maximum 7 Range 5 Interquartile Range 3 Skewness Kurtosis Mean Lower Bound 3.56 Upper Bound % Trimmed Mean 4.39

73 Recall Exam Think Write Median 4.50 Variance Std. Deviation Minimum 3 Maximum 6 Range 3 Interquartile Range 2 Skewness Kurtosis Mean % Confidence Interval Lower Bound 2.82 for Mean Upper Bound % Trimmed Mean 3.33 Median 3.00 Variance.456 Std. Deviation.675 Minimum 2 Maximum 4 Range 2 Interquartile Range 1 Skewness Kurtosis Mean % Confidence Interval Lower Bound 5.06 for Mean Upper Bound % Trimmed Mean 5.72 Median 5.50 Variance Std. Deviation Minimum 5 Maximum 8

74 Talk Range 3 Interquartile Range 1 Skewness Kurtosis Mean % Confidence Interval Lower Bound 3.39 for Mean Upper Bound % Trimmed Mean 4.22 Median 4.00 Variance Std. Deviation Minimum 2 Maximum 6 Range 4 Interquartile Range 1 Skewness Kurtosis Application Exam Recall Exam a. Lilliefors Significance Correction Tests of Normality Study Strategy Kolmogorov-Smirnov a Shapiro-Wilk Groups Statistic df Sig. Statistic df Sig. Think Write * Talk * Think Write Talk *. This is a lower bound of the true significance.

75 Application Exam Recall Exam

76 GLM recall applicat BY group /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA=ALPHA(.05) /DESIGN= group.

77 General Linear Model Input Missing Value Handling Notes Output Created 18-Feb :43:33 Comments Data Active Dataset Filter Weight Split File N of Rows in Working Data File Definition of Missing Cases Used Syntax /Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 28/Lesson 28 Data File 1.sav DataSet1 <none> <none> <none> User-defined missing values are treated as missing. Statistics are based on all cases with valid data for all variables in the model. GLM recall applicat BY group /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA=ALPHA(.05) /DESIGN= group. 30 Resources Processor Time 0:00: Elapsed Time 0:00: [DataSet1] /Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 28/Lesson 28 Data File 1.sav

78 Between-Subjects Factors Value Label N Study Strategy Groups 1 Think 10 2 Write 10 3 Talk 10 Recall Exam Application Exam Descriptive Statistics Study Strategy Groups Mean Std. Deviation N Think Write Talk Total Think Write Talk Total Box's Test of Equality of Covariance Matrices a Box's M F df1 6 df Sig..398

79 Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. a. Design: Intercept + group Multivariate Tests c Effect Value F Hypothesis df Error df Sig. Intercept group a. Exact statistic Partial Eta Squared Pillai's Trace a Wilks' Lambda a Hotelling's Trace a Roy's Largest Root a Pillai's Trace Wilks' Lambda a Hotelling's Trace Roy's Largest Root b b. The statistic is an upper bound on F that yields a lower bound on the significance level. c. Design: Intercept + group Levene's Test of Equality of Error Variances a F df1 df2 Sig. Recall Exam Application Exam Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept + group

80 Source Corrected Model Intercept group Error Total Corrected Total Dependent Variable Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Recall Exam a Application Exam b Recall Exam Application Exam Recall Exam Application Exam Recall Exam Application Exam Recall Exam Application Exam Recall Exam Application Exam a. R Squared =.559 (Adjusted R Squared =.526) b. R Squared =.237 (Adjusted R Squared =.181) DISCRIMINANT /GROUPS=group(1 3) /VARIABLES=recall applicat /ANALYSIS ALL /PRIORS EQUAL /PLOT=COMBINED /CLASSIFY=NONMISSING POOLED. Discriminant

81 Input Missing Value Handling Notes Output Created 18-Feb :50:39 Comments Data Active Dataset Filter Weight Split File N of Rows in Working Data File Definition of Missing Cases Used Syntax /Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 28/Lesson 28 Data File 1.sav DataSet1 <none> <none> <none> User-defined missing values are treated as missing in the analysis phase. 30 In the analysis phase, cases with no user- or system-missing values for any predictor variable are used. Cases with user-, system-missing, or out-of-range values for the grouping variable are always excluded. DISCRIMINANT /GROUPS=group(1 3) /VARIABLES=recall applicat /ANALYSIS ALL /PRIORS EQUAL /PLOT=COMBINED /CLASSIFY=NONMISSING POOLED. Resources Processor Time 0:00: Elapsed Time 0:00: [DataSet1] /Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 28/Lesson 28 Data File 1.sav

82 Analysis Case Processing Summary Unweighted Cases N Percent Excluded Valid Missing or out-of-range group codes At least one missing discriminating variable Both missing or out-ofrange group codes and at least one missing discriminating variable Total 0.0 Total Group Statistics Valid N (listwise) Study Strategy Groups Unweighted Weighted Think Recall Exam Application Exam Write Talk Total Recall Exam Application Exam Recall Exam Application Exam Recall Exam Application Exam Analysis 1 Summary of Canonical Discriminant Functions

83 Eigenvalues Functio n Eigenvalue % of Variance Cumulative % Canonical Correlation a a a. First 2 canonical discriminant functions were used in the analysis. Wilks' Lambda Test of Function(s) Wilks' Lambda Chi-square df Sig. 1 through Standardized Canonical Discriminant Function Coefficients Function 1 2 Recall Exam Application Exam Structure Matrix Function 1 2 Recall Exam.997 * Application Exam *

84 Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions Variables ordered by absolute size of correlation within function. *. Largest absolute correlation between each variable and any discriminant function Functions at Group Centroids Study Function Strategy Groups 1 2 Think Write Talk Unstandardized canonical discriminant functions evaluated at group means Classification Statistics Excluded Classification Processing Summary Processed 30 Missing or out-of-range group codes At least one missing discriminating variable Used in Output Prior Probabilities for Groups

85 Study Cases Used in Analysis Strategy Groups Prior Unweighted Weighted Think Write Talk Total

86

87 Canonical Correlation manova coping followup with TSR_Beh TSR_Emo /discrim all alpha(1) /print=sig (eigen dim). Manova Input Notes Output Created 22-Apr :49:38 Comments Data Active Dataset Filter Weight Split File N of Rows in Working Data File Syntax /Volumes/Cruzer/Family Journal Article/GASS Family Journal Data.sav DataSet5 <none> <none> <none> manova coping followup with TSR_Beh TSR_Emo /discrim all alpha(1) /print=sig (eigen dim). 125 Resources Processor Time 0:00: Elapsed Time 0:00: [DataSet5] /Volumes/Cruzer/Family Journal Article/GASS Family Journal Data.sav The default error term in MANOVA has been changed from WITHIN CELLS to

88 WITHIN+RESIDUAL. Note that these are the same for all full factorial designs. * * * * * * * * * * * * * * * * * A n a l y s i s o f V a r i a n c e * * * * * * * * * * * * * * * * * 120 cases accepted. 0 cases rejected because of out-of-range factor values. 5 cases rejected because of missing data. 1 non-empty cell. 1 design will be processed * * * * * * * * * * * * * * * * * A n a l y s i s o f V a r i a n c e -- Design 1 * * * * * * * * * * * * * * * * * EFFECT.. WITHIN CELLS Regression Multivariate Tests of Significance (S = 2, M = -1/2, N = 57 ) Test Name Value Approx. F Hypoth. DF Error DF Sig. of F Pillais Hotellings Wilks Roys Note.. F statistic for WILKS' Lambda is exact Eigenvalues and Canonical Correlations Root No. Eigenvalue Pct. Cum. Pct. Canon Cor. Sq. Cor Dimension Reduction Analysis Roots Wilks L. F Hypoth. DF Error DF Sig. of F 1 TO TO

89 EFFECT.. WITHIN CELLS Regression (Cont.) Univariate F-tests with (2,117) D. F. Variable Sq. Mul. R Adj. R-sq. Hypoth. MS Error MS F Sig. of F coping followup Raw canonical coefficients for DEPENDENT variables Function No. Variable 1 2 coping followup Standardized canonical coefficients for DEPENDENT variables Function No. Variable 1 2 coping followup Correlations between DEPENDENT and canonical variables Function No. Variable 1 2 coping followup Variance in dependent variables explained by canonical variables CAN. VAR. Pct Var DEP Cum Pct DEP Pct Var COV Cum Pct COV Raw canonical coefficients for COVARIATES

90 Function No. COVARIATE 1 2 TSR_Beh TSR_EMO Standardized canonical coefficients for COVARIATES CAN. VAR. COVARIATE 1 2 TSR_Beh TSR_EMO Correlations between COVARIATES and canonical variables CAN. VAR. Covariate 1 2 TSR_Beh TSR_EMO Variance in covariates explained by canonical variables CAN. VAR. Pct Var DEP Cum Pct DEP Pct Var COV Cum Pct COV Regression analysis for WITHIN CELLS error term --- Individual Univariate.9500 confidence intervals Dependent variable.. coping COVARIATE B Beta Std. Err. t-value Sig. of t Lower - 95% CL- Upper TSR_Beh TSR_EMO Dependent variable.. followup

91 COVARIATE B Beta Std. Err. t-value Sig. of t Lower - 95% CL- Upper TSR_Beh TSR_EMO * * * * * * * * * * * * * * * * * A n a l y s i s o f V a r i a n c e -- Design 1 * * * * * * * * * * * * * * * * * EFFECT.. CONSTANT Multivariate Tests of Significance (S = 1, M = 0, N = 57 ) Test Name Value Exact F Hypoth. DF Error DF Sig. of F Pillais Hotellings Wilks Roys Note.. F statistics are exact Eigenvalues and Canonical Correlations Root No. Eigenvalue Pct. Cum. Pct. Canon Cor EFFECT.. CONSTANT (Cont.) Univariate F-tests with (1,117) D. F. Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F coping followup EFFECT.. CONSTANT (Cont.)

92 Raw discriminant function coefficients Function No. Variable 1 coping followup Standardized discriminant function coefficients Function No. Variable 1 coping followup Estimates of effects for canonical variables Canonical Variable Parameter Correlations between DEPENDENT and canonical variables Canonical Variable Variable 1 coping followup

93 ANCOVA GET FILE='/Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 27/Lesson 27 Data File 1.sav'. EXAMINE VARIABLES=days predays BY group /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Explore Input Missing Value Handling Notes Output Created 25-Feb :26:34 Comments Data Active Dataset Filter Weight Split File N of Rows in Working Data File Definition of Missing Cases Used /Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 27/Lesson 27 Data File 1.sav DataSet1 <none> <none> <none> User-defined missing values for dependent variables are treated as missing. 30 Statistics are based on cases with no missing values for any dependent variable or factor used.

94 Syntax EXAMINE VARIABLES=days predays BY group /PLOT BOXPLOT NPPLOT /COMPARE GROUP /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL. Resources Processor Time 0:00: Elapsed Time 0:00: [DataSet1] /Users/richardbalkin/Desktop/CNEP 6370/GreenSalkind/Lesson 27/Lesson 27 Data File 1.sav Vitamin C Treatment Case Processing Summary Cases Valid Missing Total Vitamin C Treatment N Percent N Percent N Percent Days with Colds: Post Placebo % 0.0% % Low Vitamin C Dose % 0.0% % High Vitamin C Dose % 0.0% % Days with Colds: Prior Placebo % 0.0% % Low Vitamin C Dose % 0.0% % High Vitamin C Dose % 0.0% %

95 Days with Colds: Post Vitamin C Treatment Placebo Low Vitamin C Dose Descriptives 95% Confidence Interval for Mean 95% Confidence Interval for Mean Statistic Std. Error Mean Lower Bound 7.77 Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum 0 Maximum 20 Range 20 Interquartile Range Skewness Kurtosis Mean Lower Bound 5.66 Upper Bound % Trimmed Mean 8.61 Median 9.50 Variance Std. Deviation Minimum 0 Maximum 13 Range 13 Interquartile Range Skewness Kurtosis High Vitamin C Mean

96 Days with Colds: Prior Placebo Low Vitamin C Dose 95% Confidence Interval for Mean Lower Bound 3.92 Upper Bound % Trimmed Mean 6.39 Median 6.50 Variance Std. Deviation Minimum 0 Maximum 13 Range 13 Interquartile Range Skewness Kurtosis Mean % Confidence Lower Bound 3.87 Interval for Mean Upper Bound % Trimmed Mean 7.94 Median 8.00 Variance Std. Deviation Minimum 0 Maximum 19 Range 19 Interquartile Range Skewness Kurtosis Mean % Confidence Lower Bound 6.31 Interval for Mean Upper Bound % Trimmed Mean Median

97 High Vitamin C Dose Variance Std. Deviation Minimum 0 Maximum 19 Range 19 Interquartile Range 10 Skewness Kurtosis Mean % Confidence Lower Bound 4.75 Interval for Mean Upper Bound % Trimmed Mean 8.50 Median 9.50 Variance Std. Deviation Minimum 0 Maximum 15 Range 15 Interquartile Range Skewness Kurtosis Tests of Normality Kolmogorov-Smirnov a Shapiro-Wilk Vitamin C Treatment Statistic df Sig. Statistic df Sig. Days with Colds: Post Placebo * Low Vitamin C Dose High Vitamin C Dose * Days with Colds: Prior Placebo *

98 Low Vitamin C Dose * High Vitamin C Dose * a. Lilliefors Significance Correction *. This is a lower bound of the true significance. Days with Colds: Post

99 Days with Colds: Prior

100 UNIANOVA days BY group WITH predays /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /CRITERIA=ALPHA(0.05) /DESIGN=group predays group*predays.