Biostats Logistic Regression SPSS

PART2

Step-by-Step Guide to Assignment 6.2

This Step-by-Step Guide reviews how to manually calculate common odds ratios (OR) and the confidence interval associated with the OR, and interpret the results.

Problem 2. Compute the common odds ratio of the association between donor’s sex and the survival status of the infant, after controlling for severity.

2a. Manually calculate a common odds ratio to test the hypothesis of no association between donor’s sex and the survival status of the infant, after the inclusion of the variable severity using the CMH test.

We begin with a multiple contingency table:

	Survival Status
Disease Severity	Donor’s Sex	Alive	Dead	Total
None	Female	13	1	14
	Male	19	2	21
Mild	Female	16	1	17
	Male	37	2	39
Moderate	Female	14	1	15
	Male	31	6	37
Severe	Female	6	1	7
	Male	14	17	31
Total		150	31	181

In this table, there are four levels of disease severity (none, mild, moderate, and severe) subdivided by each level of sex (female and male). The dependent variable is dichotomous (Survival level: alive and dead), which satisfies the assumption for logistic regression.

Within this table there are four sub-tables from which four separate odds ratios can be calculated: OR of survival by sex for no disease, OR of survival by sex for mild disease, OR of survival by sex for moderate disease, and OR of survival by sex for severe disease. For CMH, the OR should be similar. Thus, we calculate each OR to determine similarity.

We will use the “shortcut” for OR = (a * d) / (b * c)

Odd Ratios:

Survival between females and males with no disease:

= (13 * 2) / (1 * 19) = 1.368421

Survival between females and males with mild disease:

= (16 * 2) / (1 * 37) = 0.864865

Survival between females and males with moderate disease:

= (14 * 6) / (1 * 31) = 2.709677

Survival between females and males with severe disease:

= (6 * 17) / (1 * 14) = 7.285714

These are not really all in the same direction. The odds are greater for survival in females with no disease, moderate disease, and severe disease. However, the odds were lower in females with mild disease than males. The concern is that the common odds, which include all odds ratios, would mask this inverse ratio because they are all combined.

For purposes of the assignment, we proceed with the calculation of the common odds, bearing this in mind.

CMH formula for common odds ratio:

OR = Σ [ai (di / ni)] / Σ [bi (ci / ni)]

Using the data from the table:

Numerator = [13 (2 / 35)] + [16 (2 / 56)] + [14 (6 / 52)] + [6 (17 / 38)]

= (0.742857) + (0.571429) + (1.615385) + (2.684211)

= 5.61388

Denominator = [1 (19 / 35)] + [1 (37 / 56)] + [1 (31 / 52)] + [1 (14 / 38)]

= (0.542857) + (0.660714) + (0.596154) + (0. 368421)

= 2.16815

Common OR = 5.61388 / 2.16815 = 2.59

Interpretation of results

2b. Interpret the results. How does the common odds ratio differ from the simple odds ratio computed in part 1? What effect might it have on your decision from part 1 to reject or fail to reject the null hypothesis?

Based on the common odds ratio (OR), females are more than twice (OR= 2.59) as likely to survive as males after controlling for disease severity.

In the simple OR, females were more than 3 times (OR= 3.275) as likely to survive than males. However, the simple OR calculation considered only the odds of survival based on the sex of the individual and did not account for disease severity.

In the common odds ratio (OR =2.59), we account for disease severity by weighting the odds ratios according to the proportion of the sample within each specific level of disease severity. This means that disease severity is taken into account in the calculation of the odds ratio. The common OR is lower than the simple OR because disease severity plays a role in survival in addition to sex.

Importance of knowing disease severity

2c. Why is it important to know the effect of severity on the association of gender and survival?

In general, disease severity would affect the survival regardless of gender. Less severe disease would likely have a bigger chance of survival than more severe disease. Therefore when assessing the association of gender and survival, it will be important to consider the effect of disease severity. As shown in the calculations for Part 2a, gender effect on the survival is not same across the disease severity levels. Females without disease or with moderate/severe disease were more likely to survive than the male in the same disease status, males with mild disease has a bigger odd of survival than female. Without considering the severity of disease in males, simple OR will overestimate the effect of sex on survival. Controlling for disease severity, we see there is still an association between sex and survival, but the association is substantially reduced. However, ORs are very different across disease severity levels, disease severity is an effect modifier. Therefore, we should report OR for each disease severity instead of using simple OR or common OR.