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# Proper usage of the one-way, repeated measures, and factorial ANOVA

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Find three research studies of interest to you that provide proper usage of the one-way, repeated measures, and factorial ANOVA.

State the null hypothesis that was tested in each study. Interpret the F statistic for each study and make a determination regarding the null hypothesis (reject or fail to reject). Comment further on interaction or main effects when appropriate.

Lastly, why do researchers utilize an ANOVA of the t-test? Find a set of data that is suitable for an ANOVA and perform both the appropriate ANOVA and the appropriate t-test on the data.

Comment on your statistical findings? Do you agree with your initial statement regarding the preference of an ANOVA over a t-test when examining more than two groups?

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The proper usage of the one-way, repeated measures and factorial ANOVA.

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One-Way ANOVA
Research Study Design:
Let us consider four hospitals namely A, B, C, and D and compare the average cost of treatment in each of these four hospitals.
For this study, random samples of thirty discharges have been selected from each of the four hospitals and data of the actual costs for these discharges have been obtained.
[Please refer to the EXCEL attachment for the data].

Null and Alternate Hypothesis:
Null Hypothesis: All the four hospitals have the same average cost of treatment.
Alternative Hypothesis: At least one of the four hospitals has a different average cost of treatment than the rest.

Analysis Plan:
We shall use significance level, α=0.05 for the test.
The test statistic used here is one-way ANOVA.

Calculation of F-Statistic:
The F-statistic has been calculated using Excel's Data Analysis.

ANOVA: Single Factor

SUMMARY
Groups Count Sum Average Variance
A 30 199230 6640.99 21757799
B 30 187617 6253.9 24442189
C 30 281421 9380.71 23920882
D 30 171486 5716.21 15607649

ANOVA
Source of Variation SS df MS F P-value F critical
Between Groups 2.4E+08 3 8E+07 3.7333898 0.01323 2.68281
Within Groups 2.5E+09 116 2.1E+07

Total 2.7E+09 119

Interpretation of results:
As we can see, the mean costs of treatment for all the four hospitals are different. But are these differences statistically significant?
With a significance level of α=0.05, Critical F = 2.6828.
Therefore, the decision rule for rejecting the null hypothesis is given as:

According to the test result F statistic = 3.7334
Since the F statistic is greater than the critical F value, we reject the null hypothesis. From our study, we conclude that there is statistically significant evidence that at all the hospitals do not have the same average cost of treatment and at least one of the four hospitals has a significantly different average cost of treatment than the rest.

Repeated Measures ANOVA:
Research Study: Are later admissions, in the case of several readmissions for the same person in a hospital, likely to be costlier than the earlier admissions in the hospital?

Research Study Design: A random sample of twelve people, who have been readmitted in a particular hospital at least ...

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