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PB 7.9 Rank of birth and age at oncet (Biostatistics)

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The corresponding notes are on chapter 7 (Page 216) :

ftp://195.214.211.1/books/DVD-009/van_Belle_G.,_Fisher_L.D.,_Heagerty_P.J._Biostatistics._A_methodology_for_the_health_sciences_(2004)(2nd_ed.)(en)(871s).pdf

Following is the problem to solve:

PB 7.9 McKeown et al. [1952] investigate evidence that the environment is involved in infantile
pyloric stenosis. The relationship between the age at onset of the symptoms in days,
and the rank of the birth (first child, second child, etc.) was given as follows:

Age at Onset of Symptoms (Days)
Birth Rank 0- 6 7-13 14-20 21-27 28-34 35-41 ≥ 42
1 42 41 116 140 99 45 58
2 28 35 63 53 49 23 31
≥3 26 21 39 48 39 14 23

(a) Find the expected value (under independence) for cell (i = 2, j = 3). For this
cell compute (observed - expected)^2/ expected.

(b) The chi-square statistic is 13.91. What are the degrees of freedom? What can you
say about the p-value?
(c) In the paper, the authors present, the column percents, not the frequencies, as
above. Fill in the missing values in both arrays below. The arrangement is the
same as the first table.

44 42 53 58 53 55 52
29 36 29 ? 26 28 28
? ? 18 20 21 17 21

The adjusted residual p-values are:

0.076 0.036 0.780 0.042 0.863 0.636 0.041
0.667 0.041 0.551 0.035 0.710 0.874 0.734
0.084 0.734 ? 0.856 0.843 0.445 0.954
What can you conclude?

(d) The authors note that the first two weeks appear to have different patterns. They also present the data as:

Age at Onset (Days)

Birth Rank 0-13 ≥14
1 83 458
2 63 219
≥3 47 163

For this table, X2 = 8.35.
What are the degrees of freedom? What can you say about the p-value?
(e) Fill in the missing values in the adjusted residual table, p-value table, and column
percent table. Interpret the data.

Adjusted Residuals
−2.89 2.89
? ?
1.54 −1.54
p−Values
0.0039 0.0039
0.065 0.065
? ?

Column %s
43 55
33 ?
24 ?

(f) Why is it crucial to know whether prior to seeing these data the investigators had
hypothesized a difference in the parity distribution between the first two weeks
and the remainder of the time period?

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Solution Summary

The expert examines the rank of birth and age at oncet for biostatistics.

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a.) In this problem, it is asking to find the expected value for cell i = 2, j = 3. This cell is in row 2, column 3 (i represents the rows, j represents the columns). We observe that this value is 63. To find the expected value, we use the following formula: E = ((Row total)(Column total))/Grand total
That is, the total number from our row of interest, multiplied by the total number of our column of interest, divided by the total of ALL values, will give us our expected value.
We thus perform the following calculaton: ((28 + 35 + 63 + 53 + 49 + 23 + 31)(116 + 63 + 39))/1033 = 59.51, thus, our expected value is 59.51.
To compute the chi square, we take our observed value (63) and our expected value (59.51) and calculate as follows: (63 - 59.51)2/59.51 = 0.205
b.) The degrees of freedom are calculated with the following equation: (r - 1)(c - 1), with r corresponding to the number of rows and c corresponding to the number of columns. Thus, we have the following calculation: (3 - 1)(7 - 1) = 12; we have 12 degrees of freedom.
When looking at a table of the critical values of the chi-square distribution, we see that a chi-square value of 13.91 with corresponding 12 degrees of freedom has a p-value of 0.20; we thus accept the null hypothesis that the probabilities amongst all populations are equal.
c.) The missing values are: ...

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