Data Analysis and Application

Data Analysis and Application

The repeated measures ANOVA is an effective statistical tool that is used to compare at least three group means for related groups, rather than independent ones. Typically, it assesses means where the participants are similar in every group – and this occurs under various situations. For instance, it may occur when the participants in the sample studied are measured multiple times to assess if there are changes in the intervention being administered (Howitt and Cramer, 2011). Additionally, the situation may appear when the sample participants are introduced to two or more trials or conditions and the researcher wants to compare each of the trials. Also, this type of ANOVA is also depicted as correlated samples or within-subjects ANOVA (Maxwell and Delaney, 2010). This paper focuses to assess if there is difference in anxiety symptoms across five weeks after administering a new treatment method and makes use of the repeated measures ANOVA as the statistical analysis test. 

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Section 1: Context of data set

The week5.sav data gives the scores for anxiety symptoms for five weeks after administering an anxiety treatment. For instance, the scores for the first week are treated as the baseline scores and the researcher aims to identify if the scores are statistically different across the five weeks. In this case, the patients are nine (N = 9) and they are measured five times to assess the changes of anxiety symptoms following the administration of the treatment. Notably, the “anxiety symptoms” is treated as the dependent variable (DV) or the outcome variable while the independent variable (IV) is “time” which is given in weeks, however it has five related groups; Week 1-5. 

The IV in this case is the within-subjects factor. Through a visual inspection the anxiety symptoms are decreasing from Week 1 through Week 5, generally. In Week 1and 2 the scores for anxiety symptoms are relatively high but they decreases across the other weeks and this may suggest that the anxiety treatment is effective in reducing the anxiety symptoms. The anxiety symptoms scores is measured using a ratio scale while the time is measured through a nominal scale. 

Section 2: Sphericity Assumption 

The analysis assumes that the sample size (n) is small for evaluating the multivariate normality. However, given that the analysis makes use of repeated measures ANOVA, the test for sphericity assumption is vital. Sphericity is depicted as the situation where the variances of the differences between the levels of the within-subjects factor are equal (Howitt and Cramer, 2011). Ideally, if this assumption is violated, that is the variances of the differences are not equal, it may lead to distorted variance calculation hence leading to a highly inflated F-ratio. Further, the Sphericity assumption is assessed when there are at least three levels of the within-subjects factor. 

However, the chance of violating the sphericity assumption increases with additional repeated-measures factors. In the case that this assumption is violated, a multivariate or univariate analysis should be utilized to correct this situation. For instance, the repeated-measures ANOVA is suitable for correcting sphericity assumption when using the univariate approach. The main assumptions of sphericity include; independence of observations and normality within the test variables (Howell, 2011). The Mauchly’s test evaluates whether the sphericity is violated or not. 

Table 1: Mauchly’s Test 

Mauchly’s Test of Sphericitya
Measure: MEASURE_1
Within Subjects EffectMauchly’s WApprox. Chi-SquaredfSig.Epsilonb
Greenhouse-GeisserHuynh-FeldtLower-bound
Week.03122.3489.010.424.526.250
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.
a. Design: Intercept  Within Subjects Design: Week
b. 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.

As said earlier, the Mauchly’s test is usd to assess if the Sphericity assumption has been met or violated (if the variances of the differences between the related groups are equal). Based on the above table, it is clear that the Mauchyl’s W is 0.031 with a P-value of 0.010. The P-value is less than alpha level of 0.05 and therefore, it can be concluded that the Mauchyl’s test is statistically significant and the sphericity assumption is violated. The violation of the sphericity assumption can be corrected using the epsilon corrections including the lower bound, Huynh-Feldt, and the Greenhouse-Geisser estimates (Howell, 2011). 

In this case, correcting the violation of the sphericity is simply done by adjusting the degrees of freedom for the violation effect – and this is achieved through multiplying one of the sphericity estimates  by the degrees of freedom. Consequently, this makes the degrees of freedom smaller which leads to a more conservative F-ratio. As a rule of thumb, when the Greenhouse-Geisser estimate (Ɛ) > 0.75, the Huynh-Feldt estimate is used, otherwise the Ɛ estimate is used. Therefore, in the case above all the epsilon estimates are less than 0.75, and based on the above rule of thumb the Greenhouse-Geisser estimate is used (Howitt and Cramer, 2011. Considering Table 2 below, it is clear that the test of within-subjects effects, this Greenhouse-Geisser estimate (0.424) is statistically significant (F = 22.786, P = 0.0000). 

Table 2: Epsilon Estimates

Tests of Within-Subjects Effects
Measure: MEASURE_1
SourceType III Sum of SquaresdfMean SquareFSig.Partial Eta Squared
WeeksSphericity Assumed1845.9114461.47822.786.000.740
Greenhouse-Geisser1845.9111.6961088.24022.786.000.740
Huynh-Feldt1845.9112.104877.13622.786.000.740
Lower-bound1845.9111.0001845.91122.786.001.740
Error(Weeks)Sphericity Assumed648.0893220.253
Greenhouse-Geisser648.08913.57047.759
Huynh-Feldt648.08916.83638.495
Lower-bound648.0898.00081.011

Section 3: Hypothesis statement, Research question s and alpha

Given the purpose of this research, the hypothesis statement can be defined, as well as, the research question. Stated as the research question is:

Research question

Does the within-subject factor “Time” affect the efficacy of new anxiety treatment?  

Research hypothesis

Null hypothesis (H0): The anxiety mean scores are equal across the five weeks

Alternative hypothesis (H1): The anxiety mean scores are different for the five weeks

This can be shorted as 

H0: Equal Means 

H1: Differences in Means

Significance level (alpha)

The alpha is the probability of committing an error (making a false decision) when the null hypothesis is true. In the repeated-Measures ANOVA the alpha level is 0.05 or 5%. Such a figure indicates that there is a 5% probability of making a wrong decision when the H0 stated above is true. 

Section 4: Reporting the findings 

Descriptive statics 

Considering the descriptive statistics in Table 3 below, Week 1 had the highest anxiety scores for the 9 patients (M = 21.89, SD = 7.044). Week 2 had a relatively lower anxiety scores than week 1 but higher than anxiety scores for week 3 through 5 (M = 21.00, SD 10.223). Additionally, week 3 had a mean anxiety score of 10.00 (SD = 3.122 while week 4 had the lowest anxiety scores (M = 6.78, SD = 3.420). Lastly, Week 5 had a relatively small anxiety mean score of 8.89 (SD = 4.595). Still it can be seen that the highest anxiety score was reported in Week 2 while the lowest anxiety score was reported in Week 4

Table 3: Descriptive statistics for anxiety scores 

Descriptive Statistics
NMinimumMaximumSumMeanStd. DeviationVarianceSkewness
StatisticStatisticStatisticStatisticStatisticStatisticStatisticStatisticStd. Error
Week1983119721.897.04449.611-.883.717
Week2953418921.0010.223104.500-.590.717
Week396159010.003.1229.750.317.717
Week49213616.783.42011.694.364.717
Week59518808.894.59521.1111.394.717
Valid N (listwise)9

Estimated Marginal Means 

Linear model which have categorical variables such as the one in this analysis have difficulties in interpreting the categorical variables. However, this can be made easier through calculating the estimated marginal means; predicted means; least square means or the expected means (Howitt and Cramer, 2011). In the data in this study, the anxiety scores for each patient vary with time (across the weeks) and the estimated marginal means helps identifying the mean response for each of the independent variable factors. The Estimated Marginal Means are given in Table 4 below:

Table 4: Estimated Marginal Means 

Estimates
Measure: MEASURE_1
WeeksMeanStd. Error95% Confidence Interval
Lower BoundUpper Bound
121.8892.34816.47527.303
221.0003.40813.14228.858
310.0001.0417.60012.400
46.7781.1404.1499.406
58.8891.5325.35712.421

Based on the above estimates, patients in the baseline had a mean anxiety score of 21.889 (SE = 2.348), though the anxiety levels reduced significantly though the following four weeks. Ideally, in week 2 after using the anxiety treatment the mean anxiety score was 21.000 (SE = 3.408); in week 3 the mean was 10.000 (SE = 1.041); week 4 the mean was 6.778 (SE = 1.140) and in the last week the mean anxiety score was 8.889 (SE = 1.532).  The trend is visually seen in the profile plot in Figure 1 below:

Figure 1: Profile plot for estimated marginal means

Within-subjects effects 

Table 5: Within-Subjects Effects

Tests of Within-Subjects Effects
Measure: MEASURE_1
SourceType III Sum of SquaresdfMean SquareFSig.Partial Eta Squared
WeeksSphericity Assumed1845.9114461.47822.786.000.740
Greenhouse-Geisser1845.9111.6961088.24022.786.000.740
Huynh-Feldt1845.9112.104877.13622.786.000.740
Lower-bound1845.9111.0001845.91122.786.001.740
Error(Weeks)Sphericity Assumed648.0893220.253
Greenhouse-Geisser648.08913.57047.759
Huynh-Feldt648.08916.83638.495
Lower-bound648.0898.00081.011

The Mauchy’s test indicated that there was violation of the sphericity assumption (Mauchy’s W = 0.031, P = 0.010). Therefore, the violation was corrected using the Greenhouse-Geisser estimate of sphericity (ε = 0.424). The findings indicate that there was significant differences in the anxiety mean scores across the five weeks, F(1.696, 13.570) = 22.786, p = 0.000. The findings suggest that there is significant difference in the anxiety means for the 5 weeks.  In this case, the p-value (0.000) is less than alpha (0.05) hence the null hypothesis is rejected. 

Within-subjects contrasts 

The main effects of the within-subjects factor, as well as, the interactions are depicted in the within-subjects contrasts table below. Considering, this is a linear model, the F-ratio confirms that the model is statistically significant at alpha 0.05 (F (1, 8) = 37.436, p = 0.000). Therefore, it can conclude that, the anxiety symptoms were statistically different across the weeks and based on the estimated marginal means, the new treatment was effective since the symptoms are reducing with time. 

Table 6: Within-Subjects contrasts 

Tests of Within-Subjects Contrasts
Measure: MEASURE_1
SourceWeeksType III Sum of SquaresdfMean SquareFSig.Partial Eta Squared
WeeksLinear1456.04411456.04437.436.000.824
Quadratic122.0321122.03213.674.006.631
Cubic214.6781214.67816.900.003.679
Order 453.157153.1572.594.146.245
Error(Weeks)Linear311.156838.894
Quadratic71.39788.925
Cubic101.622812.703
Order 4163.914820.489

Between Subjects effects 

Table 7: Between-subjects effect

Tests of Between-Subjects Effects
Measure: MEASURE_1  Transformed Variable: Average
SourceType III Sum of SquaresdfMean SquareFSig.Partial Eta Squared
Intercept8459.75618459.75673.146.000.901
Error925.2448115.656

The table above shows that there is no equality in the means for anxiety symptoms, F (1, 8) = 73.146, p = 0.000. Hence, the H0 is rejected. 

Pairwise comparison (Week 1 Set as Baseline)

Table 8: Pairwise comparison

Pairwise Comparisons
Measure: MEASURE_1
(I) Weeks(J) WeeksMean Difference (I-J)Std. ErrorSig.b95% Confidence Interval for Differenceb
Lower BoundUpper Bound
12.8892.0441.000-6.9468.724
311.889*1.996.0034.23919.539
415.111*2.044.0017.27622.946
513.000*2.217.0044.50221.498
21-.8892.0441.000-8.7246.946
311.0002.896.053-.10022.100
414.222*2.559.0054.41424.030
512.1113.164.050-.01624.238
31-11.889*1.996.003-19.539-4.239
2-11.0002.896.053-22.100.100
43.222*.813.041.1086.337
51.111.7901.000-1.9154.137
41-15.111*2.044.001-22.946-7.276
2-14.222*2.559.005-24.030-4.414
3-3.222*.813.041-6.337-.108
5-2.1111.2301.000-6.8242.602
51-13.000*2.217.004-21.498-4.502
2-12.1113.164.050-24.238.016
3-1.111.7901.000-4.1371.915
42.1111.2301.000-2.6026.824
Based on estimated marginal means
*. The mean difference is significant at the .05 level.
b. Adjustment for multiple comparisons: Bonferroni.

Considering the above Table 8, the post-hoc tests using the Bonferroni correction reveals that the mean anxiety scores for weeks 3, 4 and 5 were statistically different from the baseline anxiety score (p < 0.05). However, week 2 was not statistically different from the baseline (p > 0.05). The table clearly identifies the Weeks that had significant differences in the mean anxiety scores. 

Section 5: Conclusions 

The repeated-measures ANOVA is a suitable tool in assessing the group mean difference for data with three or more categories in its within-subject factor (IV). In this study, the researcher aimed to find the efficacy of a new anxiety treatment on anxiety levels across five weeks. Typically, time here is a categorical variable with five trials and the efficacy of the treatment was hypothesized to be influenced by the time. Based on the study findings, time (given in weeks) had a significant influence on the efficacy of the new treatment on anxiety symptoms. The null hypothesis was rejected and concluded that the anxiety symptoms for the five weeks were statistically different – indicating that as much as the new drug suppressed the anxiety symptoms, time played a vital role in influencing the efficacy of the drug. 

Strengths and Limitations of repeated measures ANOVA 

Strengths 

  • It is appropriate in testing between-subject effects associated with categorical variables 
  • It is simple to compute 
  • It helps identify which variables are different from the others 
  • More powerful as the researcher can control variables causing variation between the subjects 
  • It is possible to evaluate effects over time as seen in this study

Limitations 

  • Order effects resulting from exposing the subjects to various treatments
  • Needs additional materials for experiment such as for the treatments for each group or subjects for every trial. 
  • May result to biasedness especially when grouping is required for analysis  

References

Maxwell, S.E. & Delaney, H.D. (2010). Designing experiments and analyzing data: A model comparison perspective. Belmont: Wadsworth

Howell, D. C. (2011). Statistical Methods for Psychology. Wadsworth Publishing.Howitt, D., & Cramer, D. (2011). Introduction to Research Methods in Psychology. (3rd ed.). (pp. 164, 179-181). Harlow, Essex: Pearson Education Limited

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