**Research questions**: Is It Better to Apply Psychotherapy Treatment to a Mental Patient At Home or in a Hospital Setting?

**Appropriateness of the chi-square test of independencefor the Research Question**

The study is going to employ the use of the chi-square test of independence. The chi-square test of independence is a non-parametric (distribution free) statistical tool designed to analyze group differences when the dependent variable is measured at a nominal level (McHugh, 2013). This test is robust with respect to the distribution of the data. Specifically, it does not require equality of variances among the study population. Unlike most non-parametric, the calculations needed to compute the chi-square provide considerable information about how each of the groups performed in the study.The chi-square test of independence is used when a research question has two nominal variables and the researcher wants to establish whether the proportions of one variable are different for different values of the other variable (McHugh, 2013). It is normally used when the sample size is extremely large (Huck, 2012).For a research study to employ the use of this statistical test there must be two nominal variables, each with two or more possible values. For this test to be used, the researcher wants to establishwhether the proportions for one variable differ among values of other variables.The chi-square of independence will allow the researcher to establish whether any difference is likely to have occurred by chance.

This research question is appropriate for this test as its characteristics fit the qualification for a research study that can use this statistical test. This research study has two variables both of which are nominal, and both of which have two possible values. That is, one nominal variable is the reactions towards a particular setting which can be negative reaction or positive reaction; while the other nominal variable is the setting which also takes two possible values; home setting or hospital setting.

**The Variables in the Research Study**

The research study wants to establish whether it is better to apply psychotherapy treatment to mental illness patient in either a home setting or a hospital setting. Therefore, data to be collected will be on the reaction to psychotherapy treatment in adults between the ages of 18-59.The two variables in this research study are the reactions towards a particular setting, and the other variable is the setting where psychotherapy session takes place. Both of these variables are categorical or nominal variables which can take two or more categories but have no intrinsic ordering. That is, the reactions towards a particular setting can either be a negative reaction or a positive reaction; while the other nominal variable is the setting which also takes two possible values; home setting or hospital setting.

**Appropriateness of the Variables for a the chi-square test of independence**

The variables of the study fit the qualification for the selected statistical test. The chi-square test of independence requires that the research study has variable which are categorical or nominal, it also requires that each of the study variable have two or more possible values. The research question has two variables both of which are nominal variables; in addition, these two variables can each assume two possible values. That is, the first variable is the patient’s reactions towards a particular setting which can take the two values; negative reaction or positive reaction.The other nominal variable is the setting of the session, which also takes two possible values; home setting or hospital setting.

**Statistical Notation and Written Explanation for the Null and Alternative Hypotheses**

The null hypothesis for a chi-square test of independence states that the relative proportions of one variable are independent of the second variable. That is, the proportions at one variable are the same for different values of the second variable. Following are the statistical notations of the null hypothesis and alternative hypothesis;

H_{0}: µ_{(x)} = µ_{(y)}

H_{1}: µ_{(x)} ≠ µ_{(y)}

Where:

µ_{(x)}:Represents the proportion of patients subjected to psychotherapy in a home setting and show a positive reaction

µ(_{y}): Represents the proportion of patients subjected to psychotherapy in a hospital setting and show a positive reaction

The null hypothesis assumes that the proportion of patients subjected to psychotherapy in a home setting and have a positive reaction is equal to the proportion of patients exposed to psychotherapy in a hospital setting. While the alternative hypothesis presupposes that that the proportion of patients subjected to psychotherapy in a home setting and have a positive reaction is not equal to the proportion of patients exposed to psychotherapy in a hospital setting.

**Types of Errors That Can Occur**

The main source of errors of the chi-square test of independence streams from its weaknesses. The limitations of the chi-square test of independence include its sample size requirements, difficulty of interpretation when there are large numbers of categories in the independent or dependent variables, and tendency of the Cramer’s V to produce relatively low correlation measures even for highly significant results (McHugh, 2013). These limitations are the main sources of wrong inferences from data obtained from the chi-square test of independence.

However, when a chi-square test of independence of a table larger than 2×2 is significant, a researcher is normally inclined to investigate the results further using post-hoc tests. According to Macdonald and Gardner (2000) pairwise comparisons with the Bonferroni corrections of the p-values works well as a test of independence.

In conclusion, the chi-square test of independence is a valuable statistical analysis tool that provides considerable information about the nature of research data. It is a power statistic that enables a researcher to test hypotheses about variables measured at the nominal level. As with all inferential statistics, the results are mot reliable when the data are collected from randomly selected subjects and when sample sizes are sufficiently large that they produce appropriate statistical power.

References

Huck, S. W. (2012). *Reading Statistics and Research* (6 Edition ed.). Columbus, OH: Allyn & Bacon.

Institute for Digital Research and Education. (2014). *What statistical analysis should I use?* Retrieved September 10, 2014, from Institute for Digital Research and Education: http://www.ats.ucla.edu/stat/mult_pkg/whatstat/

McHugh, M. L. (2013). The Chi-Square Test of Independence. *The Journal of Croatian Society of Medical Biochemistry and Laboratory Medicine, 23*(2), 143-149.

Newman, C., & Newman, I. (1994). *Conceptual Statistics for Beginners.* University Press of America.

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