Non-Parametric Models

Section B...………………………………..……………………………………………………………………………………………………………………………………………………………………..………………….....….5

Section C …..………………………………………………………………………………………………………………………………………………………………………………………………………………………….….6

Section D …..………………………………………………………………………………………………………………………………………………………………………………………………………………………....….7

Conclusion……………………………………………………………………………………………………………………………………………………………………..…………….……………………………………....…...8

References …………………………………………………………………………………………………………………………………………………………………………………………………………………….…….…...9

Abstract

Keywords: nonparametric, sampling, methods, assuptions.

Unit 6 Assignment 1

The Mann-Whitney test, also called the Wilcoxon-Mann-Whitney test (Cumming, 2014; Field, 2013), is a nonparametric and rank-based test useful in determining the differences between two groups on a continuous or ordinal dependent variable. The subsequent document is structured in four main sections using the Oxoby (2008) Offers dataset. The first section, Section A, presents a research question appropriate for the Mann-Whitney test stating their null and alternative hypotheses along with the levels of measurement. The decision tree as well as the significance level for the test are also identified. The second section, Section B, presents any missing data, outliers and assumptions. The third section, Section C, presents the relevant graphs for visualization and analysis. The Mann-Whitney test results will also be reported and interpreted identifying the null hypothesis and make a decision whether to reject the hypothesis. Finally, in the fourth section, Section D, a justification for the decision and interpretation will be provided. The third section presents a G*Power computation and analysis with an interpretation of the results and a conclusion.

Section A

This section provides information about the study and the tests to be conducted specifically including:

1. The research question, the null and the alternative hypotheses;

2. The levels of measurement used and the significance level;

3. The selection process using the Decision Tree (Field, 2013).

For the Oxoby (2008) Offers dataset, the nominal variable is music (plotted on the x-axis), while offer is scale (plotted on the y-axis), with a Confidence Level =95% and Significance = .074, the following applies:

Research Question (RQ): Was there a significant difference between offers made in listening to Bon Scott compared to those listening to Brian Johnson?

Ho: The distribution of offers made in dollars is the same across categories of background music.

H1: The distribution of offers made in dollars is not the same across categories of background music.

Therefore, offers made by people listening to Bon Scott, with a median=30, were not significantly different from offers by people listening to Brian Johnson, with a median, U= 218.50, z = 1.85, p = .074, r = .31. These results were derived from Mann-Whitney tests Table 1a.

Table 1a

The Decision Tree

In using the Decision Tree (Field, 2013), analyzing the Oxoby dataset could begin by looking at the differences between sets of measurements, in contrary to looking for association. Next, the scores are ranked. Next, it was necessary to determine the number of samples to be tested and there were two; therefore, Mann-Whitney U test could be utilized (Cumming, 2014). The Mann-Whitney U test was selected for the purpose of the study in search of the differences between offers without any expected outcomes, without any correlations, without frequencies to consider and without any matched sets of variables or repeated measures to use. This particular study is not concerned about testing for or using those factors. For these reasons, the appropriate test to be conducted is the Mann-Whitney U test (Field, 2013).

Section B

When conducting a test, outliers must be identified and addressed by removing them (Cumming, 2014). Using the box plot to remove them is an accepted method of doing so for further analysis (Field, 2013). Thus, the variables could be removed by recoding them and replotting them without the outlying values. The test of normality and homogeneity of variance will provide the testing for assumptions in this section including:

1. Identifying missing data and outliers;

2. Tests of assumptions;

3. Graphs for visual analysis; and,

4. The results and conclusions.

The predictor, Brian Scott has three outliers, scores of 1, 5 and 5. While Brian Johnson predictor has one outlier, a score of 2. The lines outside the boxplots demonstrate the outliers (Figure 1). Finally, it is important to report that there were no missing values, N=0; this is substantiated by the Case Processing Summary of the SPSS output (Table 1). Finally, Figure 2 presents the boxplots after the outliers have been removed, as recommended by Field (2013) for further analysis.

Figure 1 Figure 2

Case Processing Summary
	Offer Made ($)	Cases
		Valid	Missing	Total
		N	Percent	N	Percent	N	Percent
Background Music	1	1	100.0%	0	0.0%	1	100.0%
	2	5	100.0%	0	0.0%	5	100.0%
	3	10	100.0%	0	0.0%	10	100.0%
	4	10	100.0%	0	0.0%	10	100.0%
	5	10	100.0%	0	0.0%	10	100.0%

Table 1

Field (2013) explains that significance value of Sig. <0.05 indicates a deviation from normality. Therefore, based on the tables presented the distribution in question is significantly different from a normal distribution, it is non-normal. Therefore, Table 3 presents the Tests of Normality SPSS output while Table 3 presents the Test of Homogeneity of Variance before the outliers were removed. To report, the K-S Tests for Normality before the outliers were removed Group 2 Sig.=.001, Group 3 Sig.=.003, Group 4 Sig.=.003 and Group 5 Sig.=.000. Additionally, the Shapiro-Wilk Tests for Normality for all groups Sig.=.000. Finally, after the outliers were removed the K-S and Shapiro-Wilk Tests for Normality, Group 3 Sig.=.003, Group 4 Sig.=.000, Group 5 Sig.=.000, and all Shapiro-Wilk Tests for Normality for all groups Sig.=.000, clearly indicating that the distributions were still non-significant. Overall, before and after the outliers were removed, the K-S and Shapiro-Wilk Tests for Normality consistently indicated non-normal distribution. For these reasons, it must be noted that the SPSS output indicated that the data are not normally distributed for all groups because all of the significance values are less than .05. Therefore, we reject the null hypothesis and accept the alternate hypothesis that the data do not fit the normal distribution.

On the other hand, in conducting a Test of Homogeneity of Variance, the variance of the groups should not change as the test go through the levels (Field, 2013). This means the variance of the outcome variables (the offers made, in dollars) should be the same in each of the groups. Thus, the Levene’s Test of Homogeneity of Variance tests the null hypothesis that the variances in different groups are equal. To report, the Levene’s test is non-significant at Sig. =.089 before the outliers were removed and the value of Sig. =.963 after, also non-significant because both values Sig.>.05. Therefore, it can be concluded that the null hypothesis is correct and the variances are not significantly different. For this reason, the assumption of homogeneity has been met.

Test of Homogeneity of Variancea
	Levene Statistic	df1	df2	Sig.
Background Music	Based on Mean	2.375	3	31	.089
	Based on Median	1.782	3	31	.171
	Based on Median and with adjusted df	1.782	3	12.938	.200
	Based on trimmed mean	2.375	3	31	.089
a. Background Music is constant when Offer Made ($) = 1. It has been omitted.

Tests of Normalitya
	Offer Made ($)	Kolmogorov-Smirnovb	Shapiro-Wilk
		Statistic	df	Sig.	Statistic	df	Sig.
Background Music	2	.473	5	.001	.552	5	.000
	3	.329	10	.003	.655	10	.000
	4	.329	10	.003	.655	10	.000
	5	.433	10	.000	.594	10	.000
a. Background Music is constant when Offer Made ($) = 1. It has been omitted.
b. Lilliefors Significance Correction

Table 2 Table3

Tests of Normality
	Offer Made ($)	Kolmogorov-Smirnova	Shapiro-Wilk
		Statistic	df	Sig.	Statistic	df	Sig.
Background Music	3	.329	10	.003	.655	10	.000
	4	.353	11	.000	.649	11	.000
	5	.350	15	.000	.643	15	.000
a. Lilliefors Significance Correction

Test of Homogeneity of Variance
	Levene Statistic	df1	df2	Sig.
Background Music	Based on Mean	.038	2	33	.963
	Based on Median	.030	2	33	.971
	Based on Median and with adjusted df	.030	2	23.997	.971
	Based on trimmed mean	.038	2	33	.963

Table 4 Table 5

Finally, the following Q-Q plots, Figures 3 through 10, presents the graphs for illustrating how the dots deviated from the diagonal lines before and after the outliers were removed. Therefore, it can be concluded that there is deviation from normality.

Figure 3 Figure 4 Figure 5 Figure 6

Figure 7 Figure 8 Figure 9 Figure 10

Section C

According to Field (2013), nonparametric models require no assumptions about the data and may be used with scores that are not in any numerical sense. The Mann-Whitney Test (Field, 2013) is an example of a nonparametric testing that can show the median between 2 sets of data. This section includes:

1. Graphs for analysis;

2. Test of the null hypothesis and a report of the results;

3. An interpretation of the results; and,

4. A decision whether to reject the hypothesis or otherwise.

To report, using the Oxoby (2008) dataset, the distribution was significantly non-normal with a confidence interval= 95%, effect size= .308, U=218.5, Ws =30.54, N=36, Test Statistic=1.850, and p= .074, resulting in non-significant value>.05 between the number of listener offers made to Bon Scott with a mean rank= 15.36 than Brian Johnson, mean rank =21.64. Overall, the null hypothesis should be retained, stating that the distribution of the Offers is the same for both Bonn Scott and Brian Johnson as substantiated by the table below, Table 14:

Table 6

Section D

According to Cummings (2014), the power of a statistical test denotes the Type II Error or Beta Error probability of falsely retaining an incorrect H0. Further, statistical power depends on three classes of parameters (Saha & Jones, 2016) which are: (1) the significance level; (2) the size(s) of the sample(s); and (3) an effect size parameter defining H1 and thus indexing the degree of deviation from H0 in the underlying population. Simply, the G*Power is helpful in determining the sample size required to detect an effect of a given size, when confidence interval is known.

Overall, as the power level increases, so too did the sample sizes. The following G*Power analysis shows df= 152.69, T= 1.453, total sample size= 162 and a statistical power = .95 indicating that there is 95% probability of detecting an effect. As a result, with this statistical power, the calculated effect size=.371. According to Cummings (2014), studies should have no more than a 20% probability of making a Type II error. In other words, Cummings believed (2014) studies should have an 80% probability of detecting an effect when there is an effect there to be detected.

Figure 11

t tests - Means: Wilcoxon-Mann-Whitney test (two groups)

Options: A.R.E. method

Analysis: A priori: Compute required sample size

Input: Tail(s) = One

Parent distribution = Normal

Effect size d = 0.5

α err prob = 0.074

Power (1-β err prob) = 0.95

Allocation ratio N2/N1 = 1

Output: Noncentrality parameter δ = 3.1094473

Critical t = 1.4539949

Df = 152.6986

Sample size group 1 = 81

Sample size group 2 = 81

Total sample size = 162

Actual power = 0.9507452

Overall, the G*Power results indicate that there is a 95% probability that an effect will be detected if there is an effect and a sample size of 162 is necessary. The main limitation of the study is the relatively small current sample size of 36. However, with the homogeneity assumptions has been met with Levene’s value before outliers were removed Sig.=.089 and Sig. =.963 after the outliers were removed. On the other hand, the Q-Q plots and box plots indicated non-normality of distribution and the null hypothesis is retained:

Ho: The distribution of offers made in dollars is the same across categories of background music.

Therefore, offers made by people listening to Bon Scott, with a median=30, were not significantly different from offers by people listening to Brian Johnson and as substantiated by the G*Power value =95%, there is a 95% probability that an effect will be detected if there is an effect.

Conclusion

Thus far, the preceding discussion was structured into four sections and it presented the nonparametric and rank-based test, the Mann-Whitney test, the null and alternative hypotheses along with the levels of measurement, the decision tree as well as the significance level for the tests; the missing data, outliers and the assumptions. All the relevant graphs for visualization and analysis were also presented showing deviation from normality. Finally, the report and interpretation to retain the null hypothesis were also discussed. This was summed up by the supporting explanation of the G*Power value = .95 stating a 95% probability that an effect would be detected, if it is present

References

Cumming, G. (2014). The New Statistics: Why and How. Psychological Science, 25 1.

Faul, F., Erdfelder, E., Buchner, A., & Lang, A-G. (2009). Statistical Power Analyses Using G*Power 3.1: Tests for Correlation and Regression Analyses. Behavior Research Methods, 41(4), 1149–1160. 

Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics, 4th ed. SAGE Publications (UK).

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