Question

6.48 Coffee and Depression. Researchers conducted a study investigating the relationship between caffeinated coffee consumption and risk of depression in women. They collected data on 50,739 women free of depression symptoms at the start of the study in the year 1996, and these women were followed through 2006. The researchers used questionnaires to collect data on caffeinated coffee consumption, asked each individual about physician-diagnosed depression, and also asked about the use of antidepressants. The table below shows the distribution of incidences of depression by amount of caffeinated coffee consumption. (Source: M. Lucas et al. Coffee, caffeine, and risk of depression among women. In: Archives of internal medicine 171.17 (2011), p. 1571.)

	Caffeinated coffee consumption
	≤1 cup/week	2–6 cups/week	1 cup/day	2–3 cups/day	≥4 cups/day	Total
Yes	670	373	905	564	95	2607
No	11545	6244	16329	11726	2288	48132
Total	12215	6617	17234	12290	2383	50739

(a) What type of test is appropriate for evaluating if there is an association between coffee intake and depression?

• χ2χ2-test of independence
• Z-test for a single proportion
• χ2χ2-test of goodness of fit
• Z-test for comparing two proportions

(b) Write the hypotheses for the test you identiffed in part (a). (Use plain language, not algebraic expressions.)

H0H0:

HAHA:

(c) Calculate the overall proportion of women who do and do not suffer from depression. (Round to 3 decimal places.)

Depression:

No depression:

(d) Identify the expected count for the highlighted cell, and calculate the contribution of this cell to the test statistic, i.e. (Observed−Expected)2Expected(Observed-Expected)2Expected. (Round to 1 decimal place.)

E=E=

(O−E)2E=(O-E)2E=

(e) The test statistic turns out to be χ$2=20.93χ$2=20.93. What is the degrees of freedom and the P-value? (Rouynd to 4 decimal places.)

df =

P =

(f) What is the conclusion of the hypothesis test?

• Fail to reject H0H0 -- the data does not provide convincing evidence to suggest that caffeinated coffee consumption and depression in women are associated.
• Reject H0H0 -- the data provide convincing evidence to suggest that caffeinated coffee consumption and depression in women are associated.

(g) One of the authors of this study was quoted on the NYTimes as saying it was "too early to recommend that women load up on extra coffee" based on just this study (Source: A. O'Connor. Coffee Drinking Linked to Less Depression in Women. In: New York Times, 2011). Do you agree with this statement? Explain your reasoning by explicitly mentioning whether this is a scientific study.

Answer 1

(a)

Since the test is for associateion of two categorical variables from a single population, we need to conduct

χ2-test of independence

(b)

H0: Caffeinated coffee consumption and risk of depression in women are independent.

HA: Caffeinated coffee consumption and risk of depression in women are not independent.

(c)

Overall proportion of women who do suffer from depression = 2607 / 50739 = 0.05138

Overall proportion of women who do not suffer from depression = 48132 / 50739 = 0.94862

Depression: 0.05138

No depression: 0.94862

(d)

The expected frequencies are,

	< 1 cup/week	2-6 cup/week	1 cup/day	2-3 cups/day	> 4 cups/day	Total
Yes	(2607*12215)/50739 = 627.614	(2607*6617)/50739 = 339.9854	(2607*17234)/50739 = 885.4932	(2607*12290)/50739 = 631.4675	(2607*2383)/50739 = 122.44	2607
No	(48132*12215)/50739 =11587.39	(48132*6617)/50739 = 6277.015	(48132*17234)/50739 = 16348.51	(48132*12290)/50739= 11658.53	(48132*2383)/50739 = 2260.56	48132
Total	12215	6617	17234	12290	2383	50739

Chi Square test statistic is,

Contribution of this cell to the test statistic is,

	< 1 cup/week	2-6 cup/week	1 cup/day	2-3 cups/day	> 4 cups/day	Total
Yes	(670 - 627.614)^2 / 627.614 = 2.9	(373 - 339.9854)^2 / 339.9854 = 3.2	(905 - 885.4932)^2 / 885.4932 = 0.4	(564 - 631.4675)^2 / 631.4675 = 7.2	(95 - 122.44)^2 / 122.44 = 6.1	2607
No	(11545 - 11587.39)^2 / 11587.39 = 0.2	(6244 - 6277.015)^2 / 6277.015 = 0.2	(16329 - 16348.51)^2 / 16348.51 = 0.0	(11726 - 11658.53)^2 / 11658.53 = 0.4	(2288 - 2260.56)^2 / 2260.56 = 0.3	48132
Total	12215	6617	17234	12290	2383	50739

(e)

df = (r-1) * (c-1) = (2-1) * (5-1) = 4

P = P( > 20.93, df = 4) = 0.0003

(f)

Since p-value is less than 0.05 significance level,

Reject H0 -- the data provide convincing evidence to suggest that caffeinated coffee consumption and depression in women are associated.