Question

Question 1

Researchers are interested in testing the effectiveness of a new immunosuppressive therapy to be used post-transplant to reduce graft failure in the host. Suppose researchers investigating this new therapy are concerned about the dangerous side-effects of the drug. For instance, participants that would like to enroll in the study are not allowed to already have diabetes, as the immunosuppressive agent has been found to drastically increase fasting glucose levels, which could be very dangerous for diabetics. This means, however, that participants that enter and are treated with the new drug may develop diabetes as time goes on. Suppose the following table addresses the incidence of diabetes in the study depending on treatment group.

	Type II Diabetes	No Type II Diabetes	Total
Investigational Drug	41	27	68
Best standard of Care	12	63	75
Total	53	90	143

A) Calculate the incidence of diabetes in the group after the follow-up period.

B) Calculate the relative risk of diabetes due to the investigational drug in the study.

C) Carry out a formal chi-square test to determine if there is a significant difference in diabetes between the two groups. Write out your null and alternative hypotheses at the 0.05 alpha level.

Choose the most appropriate option

		Option 1 A) Incidence group=22.06% B) RR=4.22 C) Ho: no association between treatment groups and type II diabetes in the source population Ha: association exists between the variables. Xi-Square stat=33.977 1 df Pvalue ~0 Reject the Ho that there is no association between treatment group and type II diabetes status.
		Option 2    A) Incidence group=37.06% B) RR=3.77 C) Ho: no association between treatment groups and type II diabetes in the source population Ha: association exists between the variables. Xi-Square stat=29.997 1 df Pvalue ~0 Reject the Ho that there is no association between treatment group and type II diabetes status.
		Option 3    A) Incidence group=22.16% B) RR=1.34 C) Ho: There is an association between treatment groups and type II diabetes in the source population Ha: no association exists between the variables. Xi-Square stat=17.22 2 df Pvalue ~0 Reject the Ho that there is an association between treatment group and type II diabetes status.

Answer 1

The chi-squared test for independence is applied when you have two categorical variables from a single population. It is used to determine whether there is a significant association between the two variables.

When to Use Chi-Square Test for Independence

The test procedure described in this lesson is appropriate when the following conditions are met:

• The sampling method is simple random sampling.
• The variables under study are each categorical.
• If sample data are displayed in a contingency table, the expected frequency count for each cell of the table is at least 5.

State the Hypotheses

Suppose that Variable A has r levels, and Variable B has c levels. The null hypothesis states that knowing the level of Variable A does not help you predict the level of Variable B. That is, the variables are independent.

Ho: Variable A and Variable B are independent.

Ha: Variable A and Variable B are not independent.

In this question, we need to check if there's any association between the treatment and Type II Diabetes. Thus, in this case, the hypotheses will be given as

Ho: There is no association between treatment and the risk of Type II Diabetes

Ha: There is association between treatment and the risk of Type II Diabetes

Using sample data, find the degrees of freedom, expected frequencies, test statistic, and the P-value associated with the test statistic. The approach described in this section is illustrated in the sample problem at the end of this lesson.

• Degrees of freedom. The degrees of freedom (DF) is equal to:

  DF = (r - 1) * (c - 1)

  where r is the number of levels for one catagorical variable, and c is the number of levels for the other categorical variable.

In this question, treatment has two levels and the risk of Type II Diabetes also has two levels. Thus, degree of freedom = df= (2-1)*(2-1) = 1

Now, create a table in Excel like this

	Type II Diabetes	No Type II Diabetes	Total
Investigational Drug	41	27	68
Best standard of care	12	63	75
Total	53	90	143

Expected frequencies. The expected frequency counts are computed separately for each level of one categorical variable at each level of the other categorical variable. Compute r * c expected frequencies, according to the following formula.

Er,c = (nr * nc) / n

where Er,c is the expected frequency count for level r of Variable A and level c of Variable B, nr is the total number of sample observations at level r of Variable A, nc is the total number of sample observations at level c of Variable B, and n is the total sample size.

Now, calculate the expected frequencies using Excel.

	Type II Diabetes	No Type II Diabetes	Total
Investigational Drug	25.20	42.80	68.00
Best standard of care	27.80	47.20	75.00
Total	53.00	90.00	143.00

Test statistic. The test statistic is a chi-square random variable (Χ2) defined by the following equation.

Χ2 = Σ [ (Or,c - Er,c)2 / Er,c ]

where Or,c is the observed frequency count at level r of Variable A and level c of Variable B, and Er,c is the expected frequency count at level r of Variable A and level c of Variable B.

Now, we calculate the X2 statistic for our question = (25.2-41)2/25.2 + (27.8-12)2/27.8 + (42.8-27)2/42.8 + (47.2-63)2/47.2 = 9.902+5.831+8.978+5.287 = 29.997

Now, to find the rejection region, we need to find the critical value of X2.. Open a Chi-square table using Google.

Now, as you can see, corresponding to the df = 1, the value of Chi-square statistic = 3.84. When the calculated chi-square test is larger than the table value, the null hypothesis is rejected. Otherwise, accept the null hypothesis.

Thus, our value 29.997> 3.84. Therefore, we reject the null hypothesis.

Now, to calculate the P-value, use this link to calculate. https://stattrek.com/online-calculator/chi-square.aspx

You will see that the P-value is 0.05, which is approximately 0.

Now, to calculate the relative risk ratio, we do the following calculations:

Po = probability of Type II Diabetes with best standard of care = 12/(12+63) = 0.16

P1 = probability of Type II Diabetes with investigational drug = 41/(41+27) = 0.60

RR Ratio = P1/Po = 0.6/0.16 = 3.77

Thus, option B is the correct option.