BIOSTATS 3.4 - CH.20: COMPARING TWO PROPORTIONS

• Chapter 20 discusses comparing two proportions

Homework #7, Problem 1

• Lengths of 44 great white sharks in feet are provided
• The distribution of the data is reasonably normal with one outlier in each direction
• The outliers are not extreme and preserve the symmetry of the distribution
• It is safe to use t procedures with 44 observations
• Calculate a 95% confidence interval for the mean length of great white sharks
• Determine if there is significant evidence at the 5% level to reject the claim that great white sharks average 20 feet in length
• Before accepting any conclusions, further data is needed

Homework #7, Problem 2

• The central tail feathers of long-tailed finches (Poephila acuticauda) are a sexually dimorphic trait that may play a role in sexual selection
• Longer tail feathers in males cost energy to produce, signaling the male's excellent health
• The average lengths of the two central feathers (in millimeters) of 20 male and 21 female long-tailed finches are given
• Treat the data as Simple Random Samples (SRSs) from the population of adult long-tailed finches
• Create dotplots of both data sets to see if the use of a two-sample t procedure is appropriate
• Calculate how much longer the central tail feathers of male long-tailed finches are (on average) than those of females
• Calculate a 95% confidence interval for the difference in population mean length between the male and female adult long-tailed finches

Previous Learning Objectives

• Applying inference for a population proportion has been previously discussed
• Sample proportion p̂, large and more accurate sample confidence intervals for a proportion have been covered
• Hypothesis tests for a proportion and choosing the sample size have also been previously discussed

Learning Objectives

• The learning objectives include two-sample problems for proportions
• Examine the sampling distribution of the difference between two proportions
• Calculating large sample confidence intervals for comparing proportions
• Calculating more accurate confidence intervals for comparing proportions
• Performing hypothesis tests for comparing proportions
• Find relative risk and odds ratio

Comparing Two Independent Samples

• Comparing two treatments with two independent samples is often required
• For large enough samples, the sampling distribution is approximately Normal
• Neither p₁ nor p₂ are known

Large Sample CI for Two Proportions

• For two independent Simple Random Samples of sizes n₁ and n₂ with sample proportions of successes p̂₁ and p̂₂, an approximate level C confidence interval for p₁ - p₂ is (p̂₁ - p̂₂) ± m, where m is the margin of error
• m = zSEdiff = z√((p̂₁(1 - p̂₁) / n₁) + (p̂₂(1 - p̂₂) / n₂))
• C is the area under the standard Normal curve between -z* and z*
• This method is used when the number of successes and the number of failures are each at least 10 in each sample.

Large Sample CI Example

• Assess how much the cholesterol-lowering drug Gemfibrozil reduces heart attack risk
• Incidence of heart attack is compared over a 5-year period for 2 random samples of middle-aged men taking either the drug or a placebo
• The standard error of the difference p̂₁ - p̂₂ is calculated
• SE = √((p̂₁(1 - p̂₁) / n₁) + (p̂₂(1 - p̂₂) / n₂))
• The confidence interval is (p̂₁ - p̂₂) ± z* SE
• In a study, 56 out of 2051 men on Gemfibrozil had a heart attack (2.73%), while 84 out of 2030 men on a placebo had a heart attack (4.14%)
• So the 90% CI is (0.0414 – 0.0273) ± 1.645*0.0057 = 0.014 ± 0.009
• It is 90% certain that the percent of middle-aged men who suffer a heart attack is 0.5 to 2.3 percentage points lower when taking the cholesterol-lowering drug than when taking a placebo

“Plus Four” CI for Two Proportions

• The “plus four” method produces more accurate confidence intervals
• The method acts as if there were four additional observations: one success and one failure in each of the two samples
• The new combined sample size is n₁ + n₂ + 4, and the proportions of successes are: p̃₁ = (X₁ + 1) / (n₁ + 2) and p̃₂ = (X₂ + 1) / (n₂ + 2)
• An approximate level C confidence interval is: CI: (p̃₁ - p̃₂) ± z* √((p̃₁(1 - p̃₁) / (n₁ + 2)) + (p̃₂(1 - p̃₂) / (n₂ + 2)))
• Use this method when C is at least 90% and both sample sizes are at least 5

“Plus Four” CI Example

• Researchers compared oral health in 46 young adult males wearing a tongue piercing (TP) and a control group of 46 young adult males without tongue piercing
• They found that 38 individuals in the TP group and 26 in the control group had enamel cracks
• A question to answer is how to estimate with 95% confidence the difference between the proportions of individuals with enamel cracks among young adult males with and without TP
• One count is too low for the large sample method, so the plus-four method can be used

Hypothesis Tests for Two Proportions

• The hypothesis is to test H₀: p₁ = p₂ = p
• If H₀ is true, we are sampling twice from the same population and we can pool the information from both samples to estimate p
• The pooled sample proportion is p̂ = (total successes) / (total observations) = (count₁ + count₂) / (n₁ + n₂)
• The z-score is z = (p̂₁ - p̂₂) / √(p̂(1 - p̂)(1/n₁ + 1/n₂))
• Appropriate when all counts (successes and failures in each sample) are 5 or more.

Hypothesis Test Example

• Gastric freezing was once a treatment for ulcers
• The treatment was shown to be safe and significantly reduced ulcer pain and was widely used for years
• A randomized comparative experiment compared the outcome of gastric freezing with that of a placebo
• 28 of the 82 patients subjected to gastric freezing improved, while 30 of the 78 in the control group improved
• H₀: p(gastric freezing) = p(placebo)
• Hₐ: p(gastric freezing) > p(placebo)
• Gastric freezing was not significantly better than a placebo (P-value > 0.1), and this treatment was abandoned
• The P-value is greater than 50%
• Always use a control

Relative Risk and Odds Ratio

• In the health sciences, a given health risk in the treatment group vs. the same risk in the control group is often compared
• One measure of this is the Relative Risk Reduction (RRR), which indicates how much better off one would be relative to receiving a placebo or control treatment
• RRR = (p(control) - p(treatment)) / p(control)

Relative Risk Example

• Determine how much the cholesterol-lowering drug Gemfibrozil helps reduce the risk of heart attack
• The risk of a heart attack is compared over a 5-year period for two random samples of middle-aged men taking either the drug or a placebo
• In the drug group 56 out of 2051 had a heart attack
• In the placebo group 84 out of 2030 had a heart attack
• RRR = (p(placebo) - p(drug)) / p(placebo) = (0.0414 - 0.0273) / 0.0414 ≈ 0.34
• Gemfibrozil reduces the risk of a heart attack in middle-aged men by about 34% over a 5-year period of continuous treatment, compared with middle-aged men taking a placebo (RRR = 34%)
• The risk of a heart attack over that period is 34% smaller in the Gemfibrozil group than in the placebo group
• The Absolute Risk Reduction (ARR) is simply the absolute difference in outcome rates between the control and treatment groups: ARR = p(control) - p(treatment)
• The Number Needed to Treat (NNT) is the number of patients that would need to be treated to prevent one additional negative outcome: NNT=1/ARR
• ARR and NNT are better indicators of treatment efficacy than RRR
• ARR = p(placebo) - p(drug) = 0.0414 - 0.0273 = 0.0141
• For the group taking Gemfibrozil, the rate of heart attack was a 1.4 percentage point lower than that of the placebo group (ARR = 1.4%)
• NNT = 1/ARR = 1/0.014 ≈ 70.9
• On average, we need to treat 71 men for 5 years with Gemfibrozil to avoid 1 heart attack (NNT = 71)

Group Project

• Each group can find the approved article at Topic: Group Project Discussion (instructure.com): https://marian.instructure.com/courses/3895115/discussion_topics/22915214?module_item_id=103827792
• The Manuscript Scoring Sheet can be used to assess the article as a team
• Submission due on a specific date and presentation on specific dates (subject to change)