Application of Analysis of Variance in the Analysis of HIV/AIDS-Related Depression Cases

Analysis of Variance (ANOVA)

Analysis of variance (ANOVA) is a commonly used approach in testing of the equality of various means using variance (Derrick, 2008). This analysis of often based on a number of assumptions including: independence of the samples, equal variance in populations and that the populations from which the sample is extracted has a normal or near normal distribution (Derrick, 2008). This paper applies ANOVA analysis in development of statistical assumptions defining the situation of HIV/AIDS related psychiatric depression in Africa. The paper compares the impact different therapeutic approaches in management of psychiatric depression among patients with HIV/AIDS. Using selected variables, this paper will successfully illustrate how ANOVA analysis is applicable in analysis of such scenario. To achieve this, a null and an alternative hypothesis will be developed and tested with the help of SPSS statistical analysis tool. The null and alternative hypothesis provides a rational basis upon which conclusions are drawn.

In comparing the relationship between therapy administration and psychiatric depression amongst aids patients, the paper will seek to establish whether the means of several groups are equal as well as determine if there exist any significant differences. In general, this paper aims to illustrate the logic used in ANOVA. The null hypothesis evaluated by one way ANOVA is that the mean of two or more populations are equal (Stuttgart, 2007).

It questions whether (H0) the population means for all groups bear equality and that the differences observed are a result of variations from random sampling (Brian, 2009). When null hypothesis is not true, the alternative hypothesis (Ha) supposes that the observed differences between means of sample being evaluated are real differences in the mean of the populations. The logic applied in ANOVA in mean comparison is similar to comparison of means adopted in t-tests (Green &Salkind, 2008). The data set used in this study is based on three therapy groups. One groups is subjected to journal therapy, the other group is subjected to counseling therapy while the last group is subjected to a combination of journal and counseling therapy. The data used is obtained from the General Social Survey disk. The data measures the psychiatric depression average levels suffered by HIV/AIDS patients in respective groups prior to treatment and after treatment.

In order to determine the impact of different types of treatments in management of depression related to HIV/AIDS, researches are first developed. The null hypothesis states that there is no significant difference in means of different treatment approaches adopted in depression management among HIV/AIDS infected persons in Africa. The alternative hypothesis is just the opposite of this; it states that there is a significant difference in the means of different treatment approaches adopted in depression management among HIV/AIDS infected persons in Africa. Mathematically, the expressions are expressed as follows:

  • H0: µ1 = µ2 = µ3
  • H1: µ1 ≠ µ2 ≠ µ3

Whereby µ1, µ2, ^ µ3 are the means for journal, counseling, and a combination of journal and counseling respectively. To further evaluate the significance of the test, the statistic F value is obtained. The F value is tested with a P value of 0.05 and as such an F value less than the p value will lead to outright rejection of the hypothesis being evaluated. Given that the primary ANOVA analysis does not give the actual mean differences for the groups evaluated, this study conducts further post-hoc studies to define the group differences. Tukey B is adopted for this study.

Results

The SPSS output displays the findings established based on the criteria defined earlier in the study.

Table 1: Descriptive statistics.

Descriptives
    N Mean Std. Deviation Std. Error 95% Confidence Interval for Mean Min. Max.
    Lower Bound Upper Bound
Depression scores prior to treatment Counseling and journal therapy 20 69.30 11.581 2.590 63.88 74.72 48 87
Journal therapy only 20 65.65 10.017 2.240 60.96 70.34 49 86
Counseling only 20 70.95 10.511 2.350 66.03 75.87 53 89
Total 60 68.63 10.773 1.391 65.85 71.42 48 89
Depression scores after treatment Counseling and journal therapy 20 70.30 7.981 1.785 66.56 74.04 52 88
Journal therapy only 20 65.85 8.381 1.874 61.93 69.77 50 80
Counseling only 20 71.10 7.210 1.612 67.73 74.47 60 87
Total 60 69.08 8.081 1.043 67.00 71.17 50 88

From the research study, each dependent variable i.e. treatment approach there is an associated mean as well as a standard deviation. As earlier mentioned two scenarios are evaluated: firstly, the state prior to treatment and secondly, the state after treatment. The respective means and variances are shown in table 1 attached. The returned means, for instance after treatment administration shows that the means µ1, µ2, ^ µ3 fail to satisfy the criteria defined by the null hypothesis i.e.

H0: µ1 = µ2 = µ3

Whereby µ1, µ2, ^ µ3 are 70.30, 65.85, and 71.10 respectively. However, the null hypothesis is not immediately rejected. Rather a further evaluation for statistical significance is sought.

Table 2: Test for homogeneity of variances.

Test of Homogeneity of Variances
  Levene Statistic df1 df2 Sig.
Depression scores prior to treatment .360 2 57 .699
Depression scores after treatment .285 2 57 .753

To reject the findings of the null, the F statistic value is determined. Given the F value of.699 and.753 obtained for pre and post treatment respectively, it is sufficient to reject the null hypothesis (see table 2). The F values shows that the result are statistically significant because they are larger than the p value of.05 earlier stated. There is a strong evidence suggesting that the there is a significant difference between the means of variables being evaluated.

ANOVA
    Sum of Squares df Mean Square F Sig.
Depression scores prior to treatment Between Groups 294.233 2 147.117 1.280 .286
Within Groups 6553.700 57 114.977    
Total 6847.933 59      
Depression scores after treatment Between Groups 320.033 2 160.017 2.582 .084
Within Groups 3532.550 57 61.975    
Total 3852.583 59      

The ANOVA results further reinforce the earlier defined results. It provides the results between groups as well as within groups. In both cases, the results reveal that there are indeed differences between the groups as well as within groups.

The overall SPSS and syntax files illustrating the discussion illustrated in the study are attached hereafter:

SPSS Syntax and output

Descriptives
     
    N Mean Std. Deviation Std. Error
Depression scores prior to treatment Counseling and journal therapy 20 69.30 11.581 2.590
Journal therapy only 20 65.65 10.017 2.240
Counseling only 20 70.95 10.511 2.350
Total 60 68.63 10.773 1.391
Depression scores after treatment Counseling and journal therapy 20 70.30 7.981 1.785
Journal therapy only 20 65.85 8.381 1.874
Counseling only 20 71.10 7.210 1.612
Total 60 69.08 8.081 1.043
Test of Homogeneity of Variances
  Levene Statistic df1 df2 Sig.
Depression scores prior to treatment .360 2 57 .699
Depression scores after treatment .285 2 57 .753
ANOVA
    Sum of Squares df Mean Square
Depression scores prior to treatment Between Groups 294.233 2 147.117
Within Groups 6553.700 57 114.977
Total 6847.933 59  
Depression scores after treatment Between Groups 320.033 2 160.017
Within Groups 3532.550 57 61.975
Total 3852.583 59  

Post Hoc Tests

Multiple Comparisons
Dependent Variable (I) Therapy type (J) Therapy type Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval
Lower Bound Upper Bound    
Depression scores prior to treatment Tukey HSD Counseling and journal therapy Journal therapy only 3.650 3.391 .532 -4.51 11.81
Counseling only -1.650 3.391 .878 -9.81 6.51
Journal therapy only Counseling and journal therapy -3.650 3.391 .532 -11.81 4.51
Counseling only -5.300 3.391 .270 -13.46 2.86
Counseling only Counseling and journal therapy 1.650 3.391 .878 -6.51 9.81
Journal therapy only 5.300 3.391 .270 -2.86 13.46
Dunnett t (2-sided)a Counseling and journal therapy Counseling only -1.650 3.391 .843 -9.34 6.04
Journal therapy only Counseling only -5.300 3.391 .213 -12.99 2.39
Depression scores after treatment Tukey HSD Counseling and journal therapy Journal therapy only 4.450 2.489 .183 -1.54 10.44
Counseling only -.800 2.489 .945 -6.79 5.19
Journal therapy only Counseling and journal therapy -4.450 2.489 .183 -10.44 1.54
Counseling only -5.250 2.489 .097 -11.24 .74
Counseling only Counseling and journal therapy .800 2.489 .945 -5.19 6.79
Journal therapy only 5.250 2.489 .097 -.74 11.24
Dunnett t (2-sided)a Counseling and journal therapy Counseling only -.800 2.489 .928 -6.45 4.85
Journal therapy only Counseling only -5.250 2.489 .072 -10.90 .40
a. Dunnett t-tests treat one group as a control, and compare all other groups against it.

Homogeneous Subsets

Depression scores prior to treatment
  Therapy type   Subset for alpha = 0.05
  N 1
TukeyHSDa Journal therapy only 20 65.65
Counseling and journal therapy 20 69.30
Counseling only 20 70.95
Sig.   .270
Means for groups in homogeneous subsets are displayed.
a. Uses Harmonic Mean Sample Size = 20.000.
Depression scores after treatment
  Therapy type   Subset for alpha = 0.05
  N 1
TukeyHSDa Journal therapy only 20 65.85
Counseling and journal therapy 20 70.30
Counseling only 20 71.10
Sig.   .097
Means for groups in homogeneous subsets are displayed.
a. Uses Harmonic Mean Sample Size = 20.000.

References

Brian, S. (2009). Introduction to Statistics. London: McGraw Hill.

Derrick, A. (2008). Research methods applicable to quantitative analysis of data (2th ed.). Worth publishers: New York.

Green, S.B. & Salkind, N.J. (2008) Using SPSS for Windows and Macintosh: Analyzing and Understanding Data (5th ed.) Pearson Prentice Hall: New Jersey.

Stuttgart, W. (2007). ANOVA application to case analysis.Journal of Statistics 14(2), pp. 123-126.

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StudyCorgi. 2022. "Application of Analysis of Variance in the Analysis of HIV/AIDS-Related Depression Cases." March 8, 2022. https://studycorgi.com/application-of-analysis-of-variance-in-the-analysis-of-hiv-aids-related-depression-cases/.

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