Confidence interval and hypothesis testing examples

Transcription

1 Confidence interval and hypothesis testing examples Eric F. Lock UMN Division of Biostatistics, SPH 11/20/2018

2 ICU Data Data for a random sample of n = 200 patients admitted to intensive care unit (ICU) We will consider various inference questions based on this sample. General approach: Formalize question using statistical notation and terminology (Plot or summarize the relevant data!) Obtain inference results (confidence interval, p-value) Interpret results in the context of the original question

3 Sex of ICU patients Consider the sex of ICU patients Are the proportion of males and females admitted to the ICU equal? In our sample, ˆp = 0.38 is the proportion of female patients Formalize the question using statistical notation and terminology Obtain inference results Interpret results in the context of the original question

4 Sex of ICU patients The p-value is Possible interpretations or conclusions (right, wrong)? It is unlikely that we would observe less than 38% of females in our sample if males and females are admitted in equal proportions. Because the p-value us small, there is a large difference between the rates of male and females. We have statistically significant evidence that males are admitted more frequently to the ICU. Males are admitted more frequently to the ICU.

5 Sex of ICU patients What are plausible values for the proportion of patients admitted to the ICU who are female? What are plausible values for the ratio of males to females admitted to the ICU?

6 Heart rate of ICU patients What is the mean heart rate of patients admitted to the ICU? Sample data: hist(heartrate) Histogram of HeartRate Frequency HeartRate Sample mean: x = 98.9 Find an 80% confidence interval for the population mean.

7 Heart rate of ICU patients The 80% confidence interval is (96.5, 101.3). Possible interpretations (right/wrong)? We are 80% confident that the mean heart rate of American adults is between 96.5 and beats per minute. There is a 80% chance that a randomly selected ICU patient will have a heart rate between 96.5 and beats per minute There is a 80% chance that if we randomly select a sample of 200 ICU patients, they will have an average heart rate between 96.5 and beats per minute The probability that the mean heart rate for ICU patients is between 96.5 and is 0.8.

8 Heart rate of ICU patients Let a be the 10th percentile and b be the 90th percentile of the heart rate for ICU patients Can estimate a and b from their empirical quantiles: 10% of observed values are below â, 90% of observed values are below ˆb How precise are our estimates for a and b? Create a 90% confidence interval for a, and a 90% confidence interval for b

9 Age and survival status Are age and survival status associated for ICU patients? boxplot(age Status) In our sample, patients that did not survive were years older on average Find the p-value using a resampling approach

10 Age and survival status Compute the p-value using a normal approximation H 0 : µ 1 = µ 2 If n 1 and n 2 are both large, x 1 x 2 Normal(0, 1) s1 2/n 1 + s2 2/n 2 Or, may use t-distribution instead of N(0, 1) if n 1 or n 2 are small and the data in each group are approximately normal (Devore & Beck 10.2)

11 Age and survival status Find a 95% confidence interval for the mean difference in age between those that did and did not survive.

12 Infection and survival status Do patients with an infection have poorer survival outcomes? survived not survived infection no infection Under H 0 : p 1 = p 2 = p, the CLT gives ˆp 1 ˆp 2 p(1 p)/n1 + p(1 p)/n 2 Normal(0, 1). Estimate p with the overall sample proportion: ˆp = n 1ˆp 1 + n 2ˆp 2 n 1 + n 2

13 Infection and survival status Find a 95% confidence interval for the difference in the proportion who survive with an infection and proportion who survive without an infection.

14 Age and heart rate Consider the correlation between age (yrs) and heart rate (bpm) plot(age,heartrate) HeartRate Age Find a 95% confidence interval for the correlation between age and heart rate.