Data Interpretation Practicum

Application: Data Interpretation Practicum

To investigate the research question, the study formulated the following hypothesis:

            H0: The rate of occurrence of risks is the same for the employees under female and male supervisors in the three manufacturing locations.

            H1: The rate of risks occurrence is different for the employees under female and male supervisors in the three manufacturing locations.

From the available data, the rate of risk occurrence can be tested through the analysis of 3 variables which are; injury rate, safety climate and risk. A Univariate 2-way ANOVA will be conducted using SPSS at a 95% confidence interval.

From the above hypothesis, we are able to formulate the hypothesis to be tested by the 2 way ANOVA test;

H0: The rate of occurrence of risks is the same for the three manufacturing locations.

            H1: The rate of risks occurrence is the same for the employees under female and male supervisors

            H3: The 2 factors (manufacturing location & supervisor gender) are independent or the interaction effect does not exist.

The F-statistic is computed for each hypothesis based on the dependant variable under test.

Assumptions of 2-way ANOVA

1. The dependent variable should be distributed normally for each population. The cells of the design should define the different populations i.e. the combinations of the levels of the 2 factors (Wilcox 2001). He states that a 3 by 2 ANOVA has six cells and the assumption states that the dependant variable should be normally distributed for the six cells. The sample size is required to be relatively large so as to produce valid p values.
2. Population variances of the dependant variables should be equal for all cells. According to Wilcox 2001, sample sizes for the dependent variable may differ hence violating this assumption which may render the p values untrustworthy.
3. Independence of scores. According to Wilcox 2001, the observations of the dependent variable represent random samples from the populations and the observations are independent of each other. In case this assumption is violated, the two-way ANOVA yields inaccurate p values.

From the above assumptions, it is evident that the sample sizes for the dependent variables are not equal hence violating the second assumption that population variances should be equal. Due to these circumstances, we shall use statistics which do not assume the equality of population variances such as the Browne-Forsythe or the Welch statistics. For post hoc tests, the validity of the results is questionable if the population variances differ. If the variances are different, we choose one of the methods which doesn’t assume equality of population variances such as the Dunnett’s procedure (Hotchberg and Tamhane, 1987).

Conducting tests of main and interaction effects

To conduct these tests, we first define our variable where the main effect variables are manufacturing location and supervisor gender while the interaction effect is manufacturing location*supervisor gender as shown in the table below.

Variable	Definition
Manufacturing location   Gender of child	1 = Boston 2 = Phoenix 3 = Seattle     1 = Male 2 = Female
Injury rate	Measuring index

Table 1

According to Kumar (2008), the following steps were followed to conduct the test. From the data editor menu bar, click on analyze then a drop down menu pops and general linear model is chosen then Multivariate. In the Univariate dialog box, choose injury rate then move it to the dependant variable box. Hold the control key then choose manufacturing location and supervisor gender then move them to the fixed factor(s) box. Click options and from the dialog box which appears, press the control key then choose supervisor gender, manufacturing location and supervisor gender*manufacturing location on the factor(s) and factor interactions box and move them to the display means for box. In the display box, click homogeneity tests, estimates of effect size, and descriptive statistics. In the post hoc box choose Dunnett’s C test under the equal variances not assumed radio button.

INTERPRETATION

Table 3 from the appendix provides the descriptive statistics obtained after conducting a 2-way ANOVA test. From the table, an examination of the means shows that the ordering of the means for the three manufacturing locations is the same for both genders.

Tests of between-subjects effects are shown in table 4 in the appendix. From the table, a three by two ANOVA was carried out to investigate the effects of manufacturing location and supervisor gender on occurrence of risk (injury rate) at a significant p value where α is less than 0.05. The two-way ANOVA test indicated that no significant interaction existed between manufacturing location and supervisor gender, F (2, 45) = 0.463, p = 0.632, partial n² =0.020. The ANOVA test also indicated that there was no significant main effect for manufacturing location, F (2, 45) = 0.075, p = 0.928, partial n² =0.003. In addition, there was also no significant main effect for supervisor gender, F (1, 45) = 0.145, p = 0.705, partial n² =0.003 (Gay, et al, 2009). The F-statistic shows that neither significant main effects nor interaction effects existed between the 2 factors under study hence there was reduced need for further study.

Since we not have a statistically significant interaction effect, we interpret the post hoc results of the different manufacturing towns which are found in the multiple comparisons table which is table 5. From the table, the Turkey post hoc test shows that there is no statistically significant difference of means between manufacturing locations i.e. Boston and Phoenix, Boston and Seattle, Seattle and Phoenix at (p < .0005) since all the significance values from the table are greater than 0.0005. This tests our first hypothesis where we accept the null hypothesis and therefore conclude that the rate of occurrence of risks is the same for the three manufacturing locations.

The F-Statistic for the interaction effect between manufacturing locations and supervisor gender as indicated above is, F (2, 45) = 0.463, p = 0.632. The p value shows no statistically significant interaction between the 2 independent fixed factors. This helps in testing our third hypothesis where we accept the null hypothesis since the p value is greater than 0.05 and eventually conclude that the factors (manufacturing locations and supervisor gender) are independent and no interaction effect exists.

In addition, the F-Statistic for the supervisor gender represented by F (1, 45) = 0.145, p = 0.705 shows that there was no statistically significant main effect. This helps in testing the second hypothesis where we accept the null hypothesis since the p value is greater than 0.05. We therefore conclude that the rate of risks occurrence is the same for the employees under female and male supervisors.

ANOVA test was the best choice for this study due to the presence of several fixed factors which influence the dependent variables as opposed to t-tests which can handle only a single fixed factor or independent variable (Wilcox 2001).

The box plot below shows the distributions of injury rate between manufacturing locations and the supervisor gender. The distribution of the scores achieves our assumption of normality in the scores.

To create the box plot, click graphs from the data editor menu bar. Then choose legacy dialogs and click on box plots from the drop down menu which appears. Choose clustered then define the variables where the injury rate is moved to the variable box, manufacturing location is moved to the category axis box and supervisor gender is moved to define clusters by box then click ok (Kumar, 2008).

Figure 1: (Source: SPSS output)

References

Gay, L,R., Mills, E. G., & Airasian, P.,(2009). Educational Research: Competencies for    Analysis            and Applications (10^th ed.)

Hochberg, Y., & Tamhane, A. (1987). Pairwise comparisons. In Multiple Comparison      Procedures. Newyork: Wiley series.

Kumar, R. (2008). Research Methodology: A step-by-step Guide for Beginners. Greater   Kalash:            Sage Publications

Wilcox, R. (2001). Substantially Improving Power and Accuracy. In Fundamentals of       Modern Statistical Methods . London: Springer Science & Business Media.

APPENDICES

Table 2

Between-Subjects Factors
		Value Label	N
manufacturing location	1	Boston	15
2	Phoenix	19
3	Seattle	17
Supervisor Gender	1	Male	24
2	Female	27

Table 3

Descriptive Statistics
Dependent Variable: Injury Rate
manufacturing location	Supervisor Gender	Mean	Std. Deviation	N
Boston	Male	1.743784E1	15.2492323	10
Female	1.201363E1	11.2373506	5
Total	1.562977E1	13.8771076	15
Phoenix	Male	1.311005E1	10.6337983	4
Female	1.826225E1	23.4075637	15
Total	1.717757E1	21.2051569	19
Seattle	Male	9.797115E0	11.3287155	10
Female	1.645270E1	22.1484514	7
Total	1.253765E1	16.3569176	17
Total	Male	1.353291E1	12.9881295	24
Female	1.663595E1	20.8160156	27
Total	1.517570E1	17.4746773	51

Table 4

Tests of Between-Subjects Effects
Dependent Variable: Injury Rate
Source	Type III Sum of Squares	df	Mean Square	F	Sig.	Partial Eta Squared
Corrected Model	561.842a	5	112.368	.344	.884	.037
Intercept	8820.941	1	8820.941	26.991	.000	.375
Site	49.085	2	24.542	.075	.928	.003
SupervisorGender	47.410	1	47.410	.145	.705	.003
Site * SupervisorGender	302.550	2	151.275	.463	.632	.020
Error	14706.375	45	326.808
Total	27013.607	51
Corrected Total	15268.217	50
a. R Squared = .037 (Adjusted R Squared = -.070)

Table 5

Multiple Comparisons
Dependent Variable: Injury Rate
	(I) manufacturing location	(J) manufacturing location	Mean Difference (I-J)	Std. Error	Sig.	95% Confidence Interval
	Lower Bound	Upper Bound
Tukey HSD	Boston	Phoenix	-1.547804	6.2440100E0	.967	-16.680872	13.585263
Seattle	3.092121	6.4040020E0	.880	-12.428706	18.612947
Phoenix	Boston	1.547804	6.2440100E0	.967	-13.585263	16.680872
Seattle	4.639925	6.0352679E0	.724	-9.987232	19.267082
Seattle	Boston	-3.092121	6.4040020E0	.880	-18.612947	12.428706
Phoenix	-4.639925	6.0352679E0	.724	-19.267082	9.987232
Dunnett C	Boston	Phoenix	-1.547804	6.0418971E0		-17.106078	14.010469
Seattle	3.092121	5.3456933E0		-10.790275	16.974517
Phoenix	Boston	1.547804	6.0418971E0		-14.010469	17.106078
Seattle	4.639925	6.2772931E0		-11.451372	20.731222
Seattle	Boston	-3.092121	5.3456933E0		-16.974517	10.790275
Phoenix	-4.639925	6.2772931E0		-20.731222	11.451372
Based on observed means.  The error term is Mean Square(Error) = 326.808.