Analysis of Variance, Sign, and Wilcoxon Tests: Statistical Tests

This assignment provides an analysis of three key statistical tests: Analysis of Variance (ANOVA), Sign Test, and Wilcoxon Test. It begins by outlining the assumptions underlying the ANOVA F-test, including the random selection of samples, normal distribution of treatment populations, homogeneity of variances, and additive nature of effects. The assignment then presents a practical example involving the effect of soya protein levels on different food products, detailing the formulation of null and alternate hypotheses and the interpretation of the Fisher-F value. The document also explores the Sign test, a non-parametric test used for non-normal data, discussing its assumptions and illustrating its application with an example comparing psychology scores. Finally, it examines the Wilcoxon test, another non-parametric method used as an alternative to the t-test, particularly when data does not follow a normal distribution. The assignment covers the test's assumptions and provides an example involving the comparison of gambling playing intentions. Each test explanation includes the null and alternate hypotheses, and the interpretation of p-values. The assignment concludes with a reference to the sources used.

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Statistical Tests
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Analysis of Variance test
 Assumptions of ANOVA F –test are as follows
I. Samples are selected in a random manner from treatment populations.
II. All the treatment populations follow Gaussian (Normal) distribution.
III. All the treatment populations have same variances and are homogeneous in
nature.
IV. There is no correlation between the mean and variances of the samples.
V. The primary effects are additive in nature, that is block and treatment effects are
additive.
VI. The residuals follow Gaussian distribution and are independent in nature.
 Null and Alternate Hypotheses for ANOVA are as follows
Example: The effect of soya protein level of three separate foods is being found.
Data for the test: Quantity of soya protein levels for 1122 samples. The response factor is
Soya Protein Level (%), where the factor is FOOD type and measurement levels are 3
(vitamin A, B, and D).
Null hypothesis: H0: There is no significant effect of soya protein level due to different
food products ( μ1=μ2=μ3 ).
Alternate hypothesis: HA: There is a significant effect of soya protein level due to different
food products.
The Fisher-F value is found as described in the following table (Meyers, Gamst &Guarino,
2016).

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Figure 1: ANOVA Table
If the F value falls in the critical region corresponding to p-value as less than 0.05, then the
null hypothesis gets rejected. Otherwise, null hypothesis fails to get rejected.
 The validity of the assumptions is then cross-checked.
I. Independence of the sample gets checked from the measures of ANOVA. If repeated
measures are present then samples are not potentially independent.
II. For checking normality of the sample data, Shapiro-Wilk test is performed along
with Box-plot construction.
III. Variances of the samples are cross-checked for equality and assumption of variance
homogeneity is established.

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Sign Test
The non-parametric Sign test associated with the sign of the data which the test assigns
from the hypothesized mean value. The comparison between the clusters dimension is
done in Sign test. The Sign test is generally used for non-normal distribution data.
 Assumptions of the Sign test are as follows,
I. The compared groups should be from two different samples
II. The number of observations for both the samples should be equal.
III. The two samples should have paired data to be compared.
IV. The differences should be independent of each other.
Example: Comparison of scores in Psychology for students of the university is compared
for 32 students for spring and fall seasons. Is there any improvement in scores?
The median of the scores is found for both the data sets.
 The null and alternate hypotheses are as below,
Null hypothesis: H0: There is no difference in median scores for spring and fall seasons.
Alternate hypothesis: HA: The difference in median scores for the two seasons is not
equal to zero.
The numbers of positive and negative signs of the difference of Psychology scores
between the two seasons are found. The significance value or p-value is found by using
the Binomial distribution. The null hypothesis gets rejected for the p-value less than 0.05,
otherwise for p greater than 0.05 null hypothesis gets failed to be rejected.
The Shapiro-Wilk test and Box-plot of the two set of data are checked for nature of data
and validity of the method.

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Wilcoxon Test
This test is nonparametric and used as an alternative to the two-sample t-test. If the population
distribution does not follow the normal distribution, the Wilcoxon rank sum test is very useful in
comparing two independent samples. For comparing two samples for repeated measures,
Wilcoxon sign test is used.
 Assumptions of Wilcoxon test are as follows,
I. The samples should be independent in nature that is paired sample data are independently
drawn in a random manner.
II. As Wilcoxon Sign test checks the data for a pre and a post situation, the sample data
needs to be dependent on each other.
III. The measurement of the dependent variables should be ideally ordinal or in ratio scale.
IV. The variables of the test should have continuity for the data set.
Example: Comparison of gambling playing intentions by gambling scores of the control
group and experimental group where the experimental group is expected to be lured with
near win situations is a case for Wilcoxon test.
The null and alternate hypotheses for the test are as follows,
Null hypothesis: H0: The control and experimental group have no difference in gambling
playing intentions based on average gambling scores.
Alternate hypothesis: HA: There is a significant difference in gambling playing intentions
for both the groups.

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The validity of the test is established by the test statistic, W is calculated from the sum of the
sign of ranks of the sample data. The z-score is found by the ratio of W and standard
deviation of W. the p-value signifies the acceptance of the result. If p-value less than 0.05,
the null hypothesis gets rejected. Otherwise, for p-value greater than 0.05, the null hypothesis
fails to get rejected (Bethea, 2018). In post hoc analysis using the Bonferroni test, the validity
of the data is checked.
References
Bethea, R.M., 2018. Statistical methods for engineers and scientists. Routledge.
Meyers, L.S., Gamst, G. and Guarino, A.J., 2016. Applied multivariate research: Design and
interpretation. Sage publications.