A low birth weight dataset records 100 births and whether or not these babies experiences a germinal matrix hemorrhage (grmhem). The babies’ 5 minute apgar score is recorded (apgar5) and also whether or not the mother had toxemia (tox) during her pregnancy.
The odds ratio of whether a baby experienced a germinal matrix hemorrhage associated with a 1 unit increase in apgar score is exp(β1) = exp(-0.2496) = 0.7791124. To understand this better I referred to an article on the medical website healthline.com which states, “A score of 7 to 10 after five minutes is ‘reassuring.’ A score of 4 to 6 is ‘moderately abnormal. A score of 0 to 3 is concerning.” The coefficient indicates a negative correlation; this means that as the scores increase, the baby is less likely to experience a hemorrhage. According to statology.org, we can the use following formula to quantify the change in odds: (OR-1) * 100. Therefore, (0.779107–1) * 100 = -22.0893; with a 1 unit increase in apgar score, the baby is 22% less likely to experience a hemorrhage.
If a child has a 5 minute apgar score of 3, the predicted probability that this child will experience a brain hemorrhage is:
A child with an apgar score of 3 has a 25.87% probability of experiencing a brain hemorrhage.
A three unit increase: exp(3 * -0.2496) = 0.4729337; each 3 unit increase is associated with an increment in the odds of a hemorrhage by a factor of .47. Using the statology formula, 0.4729337–1 = 0.5270663, this means that with a 3 unit increase the patient is 52.7% less likely to experience a hemorrhage.
The odds ratio of whether a baby experienced a germinal matrix hemorrhage associated with a 1 unit increase in their tox score is exp(β2) = exp(−1.4604) = 0.2321434. This is the estimated odds of suffering a germinal matrix hemorrhage for children whose mothers were diagnosed with toxemia relative to children whose mothers who were not diagnosed. Intriguingly, the coefficient indicates a negative correlation. This means that if toxemia is present in the child’s mother (indicated by a score of 1), the baby is actually less likely to experience germinal matrix hemorrhage. If we calculate as we did before, (0.2321434–1)*100 =-76.78566, we find that the babies are 76% less likely to experience germinal matrix hemorrhage given that their mothers were diagnosed with toxemia.
If a mother was diagnosed with toxemia during pregnancy, the predicted probability of her baby experiencing a germinal matrix hemorrhage is:
The Wald chi-square statistic for the model is 5.7209 with a p-value of 0.01676. This test follows a chi-square distribution. We reject the null hypothesis that β1 = 0.
The Wald 95% Confidence Interval:
0.2496 — (1.96*0.1044), -0.2496 + (1.96*0.1044)
= (-0.454224, -0.044976)
Wald 95% Confidence Interval for the Odds Ratio:
exp(-0.454224), exp(-0.044976) =(0.6349405, 0.9560204)
This odds ratio CI does not include 1. This is evidence that the groups are significantly statistically different. This is evidence that apgar5 is significantly associated with grmhem.
An additional logistic regression model shows that there is no significant association between toxemia and germinal matrix hemorrhage. We fail to reject the null hypothesis that β2 = 0.
Patient death following icu admission was studied. There were 2500 subjects. The variables included vital status, sta (0 = dead, 1 = alive), age, sex (0=male, 1=female), race (race=white, race2=black, race3=other; I made these into dummy variables), and chronic renal failure, crn, (0=no, 1=yes).
A hypothesis test of β1 = 0 is rejected. We fail to reject the claim that Chronic Renal Failure is not significantly associated with death in the ICU.
What is the estimated odds ratio of death associated with crn for an individual age 30, female, and black and the estimated odds ratio of death associated with crn for an individual age 50, male, and white?
There are no interaction terms related to crn, and in logistic regression the other predictors are treated as constants, therefore, the odds ratio of death associated with crn does not change given the other predictor’s values. The odds ratio of death for both of these individuals is exp(β2) = exp(1.32132) = 3.748.
The odds ratio for every 10 years increase in age is exp(β1*10)= exp(0.01914*10) = 1.21.
The estimated probability of death of a 50 year old, black, female with crn:
What is the estimated odds ratio of death associated with crn for an individual age 30, female, and black and the estimated odds ratio of death associated with crn for an individual age 50, male, and white?
In this model, crn interacts with sex, therefore, the odds ratio of death depends on sex; all other values are held constant. For an individual age 30, female, and black, the estimated odds ratio of death is exp(β3 + β6) = exp(1.069531 + 0.409390) = 4.388208. For an individual age 50, male, and white, the odds ratio is exp(β3) = exp(1.069531) = 2.914013.
Test whether sex is significantly associated with sta among people with a history of crn.
Hypothesis: β2 + β6 = 0 vs β2 + β6 /= 0
Likelihood Ratio Test:
Model 1: icu$sta ~ icu$age + icu$sex + icu$crn + icu$race2 + icu$race3 +
icu$crn * icu$sex
Model 2: icu$sta ~ icu$age + icu$crn + icu$race2 + icu$race3
DF Model 1: 7 LogLikelihood: -1472.2
DF Model 2: 5 LogLikelihood: -1497.7
ChiSq: 51.127
P-value: 7.907e-12
Conclusion: Reject the Null Hypothesis
The data supports the claim that sex is significantly associated with sta among people with a history of crn.
By Aspen Gulley on .
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