Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

Model 1:AR1 correlation structure

Variable Coefficient

Empirical

Std Err

Wald

p-value

INTERCEPT 1.3978 1.1960 0.2425
BIRTHWGT 0.0005 0.0003 0.1080
GENDER 0.0024 0.5546 0.9965
DIARRHEA 0.2214 0.8558 0.7958

Effect of DIARRHEA:

dOR¼expð 0 : 2214 Þ¼ 1 : 25

95 %CI ¼exp½ 0 : 2214  1 : 96 ð 0 : 8558 ފ

¼ð 0 : 23 ; 6 : 68 Þ

Working correlation matrix: 9 9

AR1 working correlation matrix

(99 matrix: only three columns shown)
COL1 COL2 ... COL9

ROW1 1.0000 0.5254 ... 0.0058
ROW2 0.5254 1.0000 ... 0.0110
ROW3 0.2760 0.5254 ... 0.0210
ROW4 0.1450 0.2760 ... 0.0400
ROW5 0.0762 0.1450 ... 0.0762
ROW6 0.0400 0.0762 ... 0.1450
ROW7 0.0210 0.0400 ... 0.2760
ROW8 0.0110 0.0210 ... 0.5254
ROW9 0.0058 0.0110 ... 1.0000

Estimated correlations:

^r¼ 0 : 5254 for responses 1 month
apart (e.g., first and second)

^r¼ 0 : 2760 for responses 2 months
apart (e.g., first and third,
seventh and ninth)

The parameter estimates forModel 1(autore-

gressive – AR1 correlation structure) are pre-

sented on the left. Odds ratio estimates are

obtained andinterpretedin a similar manner

as in a standard logistic regression.

For example, the estimated odds ratio for the

effect of diarrhea symptoms on the outcome

(a low weight-for-height z-score) is exp

(0.2214)¼1.25. The 95% confidence interval

can be calculated as exp[0.2214  1.96

(0.8558)], yielding a confidence interval of

(0.23, 6.68).

The working correlation matrix for each of

these models contains nine rows and nine col-

umns, representing an estimate for the month-

to-month correlation between each infant’s

responses. Even though some infants did not

contribute nine responses, the fact that each

infant contributed up to nine responses

accounts for the dimensions of the working

correlation matrix.

The working correlation matrix for Model 1 is

shown on the left. We present only columns 1,

2, and 9. However, all nine columns follow the

same pattern.

The second-row, first-column entry of 0.5254

for the AR1 model is the estimate of the corre-

lation between the first and second month

measurements. Similarly, the third-row, first-

column entry of 0.2760 is the estimate of the

correlation between the first and third month

measurements, which is assumed to be the

same as the correlation betweenanytwo mea-

surements that are 2 months apart (e.g., row 7,

column 9). It is a property of the AR1 correla-

tion structure that the correlation gets weaker

as the measurements are further apart in time.

Presentation: II. Example 1: Infant Care Study 545