BIOSTATS 3.2 - CH. 18: COMPARING TWO MEANS

• Chapter focuses on comparing two means from different samples.

Learning Objectives

• Differentiate between two-sample situations.
• Understand and apply two-sample t procedures.
• Assess the robustness of these procedures.
• Know when to avoid pooled procedures.
• Avoid making inferences on standard deviations.

Two Sample Situations

• Comparison of two treatments or conditions may be needed.
• Determination of sample independence is important.
• Independent samples involve individuals chosen separately in both samples.
• Matched pairs samples involve related individuals, like subjects assessed twice or siblings.

T Distribution for Two Independent Samples

• Two independent Simple Random Samples, or SRSs, come from two populations with unknown (μ₁, σ₁) and (μ₂, σ₂).
• (X₁, s₁) and (X₂, s₂) are used to estimate (μ₁, σ₁) and (μ₂, σ₂) respectively.
• In theory, both populations should be normally distributed.
• In practice, both distributions should have similar shapes without strong outliers for the t procedures to be valid.

T Statistic

• The two-sample t statistic follows approximately a t distribution.
• Standard error (SE) in the denominator reflects variation from both samples.
• Degrees of freedom are computed with a complex formula, often done by statistical software.

Two-Sample T-Test

• For two independent random samples, the goal is to test the null hypothesis.
• Null hypothesis is: H₀: μ₁ = μ₂ ↔ μ₁ – μ₂ = 0.
• Alternate hypothesis can be one-sided or two-sided.
• Compute the t statistic and appropriate degrees of freedom (df).
• Obtain and interpret the P-value based on the alternate hypothesis.

Lung Capacity and Parental Smoking Example

• Forced Vital Capacity, or FVC, is the volume of air exhaled in 6 seconds, measured in milliliters.
• FVC was obtained from samples of children exposed and not exposed to parental smoking.
• In a sample of children exposed to parental smoking the mean FVC was 75.5 with a standard deviation of 9.3 from a sample size of 30.
• In a sample of children not exposed to parental smoking the mean FVC was 88.2 with a standard deviation of 15.1 from a sample size of 30.
• A two-sample t test was used to assess the impact of parental smoking on child lung capacity.
• The t statistic for the lung capacity test example is equal to -3.92
• The null hypothesis was rejected as the P value was less than 0.0005, based off a df of 40
• Lung capacity is significantly impaired in children exposed to parental smoking compared to those not exposed.

Hypothesis

• Geckos have specialized toe pads for climbing slick surfaces.
• Male mean toe pad area in Tokay geckos is 6.0 cm², and female is 5.3 cm².
• Need to formulate the null hypothesis for testing if male and female geckos differ significantly in toe pad size

Two Sample Confidence Interval

• Uses the difference between sample averages (X1 - X2) to estimate (μ₁ - μ₂).
• C is the area between -t* and t*.
• Find t* in Table C for the computed degrees of freedom.
• The margin of error m is calculated as m = t*√(s₁²/n₁ + s₂²/n₂) = t*SE.

Pesticide and Seedling Growth Example

• Seeds planted in pots with pesticide-treated soil vs. untreated soil.
• Seedling growth (in mm) recorded after 2 weeks.
• The degrees of freedom (df) are calculated to be 37.86.
• A 95% confidence interval for (µ₁ – µ₂) is (X1 - X2) ± t*√(s₁²/n₁ + s₂²/n₂).
• Using df = 30 from Table C, resulting in (51.48 – 41.52) ± 2.042√(11.01²/21 + 17.15²/23) equating to 9.96 ± 8.80 mm.

Software Examples

• For df = 38, m is approximately 2.024 * 4.31 ≈ 8.72.
• Software outputs provide values for t Statistic, P (T<=t) one-tail, t Critical one-tail, P (T<=t) two-tail, and t Critical two-tail.
• A 95% level of confidence using pesticide yields seeds that are 1.2 to 18.7 mm longer on average after 2 weeks.

Robustness

• Two-sample statistic is most robust with equal sample sizes and similar sample distributions.
• Two-sample tests remain quite robust even when deviating from these conditions.
• A combined sample size (n₁ + n₂) of 40 or more allows working even with the most skewed distributions.

Pooled Two-Sample T Procedures

• Two versions of two-sample t procedures: one assuming equal variance ("pooled") and one not assuming equal variance for the two populations.
• Pooled two-sample t test has degrees of freedom n₁ + n₂ – 2.

Inference and Standard Deviations

• The pooled t test assumes equal variances.
• Procedures exist to test whether two population variances are equal, such as F procedures.
• Avoid using F procedures without proper guidance due to a lack of robustness.

T Procedures Overview

• One-sample t procedure summarizes one sample by its mean X and standard deviation s to make inferences about population parameter μ when σ is unknown.
• Matched Pairs t Procedure summarizes two paired data sets (from a matched pairs design) from n pairwise differences ( ) to make inferences about population parameter
• Two-Sample t Procedure summarizes two independent samples (unrelated individuals) based on with ( to make inferences about -

Procedure Selection

• Blood pressure change with oral contraceptives: Compare women using/not using OC, a two-sample or two-sample t procedure.
• Cholesterol level in adults with "high cholesterol" parents: Average cholesterol level in general adult population is 175 mg/dL.
• Compare a sample of adults with "high cholesterol" parents, a one-sample t procedure.
• Take a sample of bread loaves, and compare the vitamin content right after baking and again 3 days later with a matched pairs sample t procedure.
• Bread loaves just baked compared to bread loaves stored for 3 days: Compare vitamin content, a two-sample t procedure.

Paired Sample T-Test

• The test deals with the previous two sample t-test with independent samples.
• Each datum in one sample is in no way associated with any specific datum in the other sample.
• Test when the test has an observation that can be correlated in some way to an observation in another sample - dependence.
• Can have dependence because the test looks at individual at repeated time points, at the same individual but different locations, or in case of a patient study when patients are paired based on age, gender, and other factors to get a better control for your study.
• The assumption is that the differences between the samples come from a normally distributed population, instead of the individual values themselves.
• Since the regular two sample t-test assumes all data are independent, a new test statistic is needed for dealing with dependent samples.

Deer Example

• Null hypothesis: the left foreleg and left hindleg lengths of deer are equal.
• Testing would entail the measurement on a number of deer involving variation among the data owing to the null hypothesis being false and the leg lengths differ.
• There would also be variation as deer are of different sizes, and a deer with longer front leg is likely to have a longer hindleg.
• Degrees of freedom are calculated as t = d/SEM = 3.3/0.97 = 3.402 vs to.05(2),9 = 2.262.
• Hypothesis is rejected, that p – value = .008.

Confidence Intervals

• Probability is that some interval around the mean or instance of interest actually contains the mean.
• Contains an interval with a lower and upper value.
• Formula:
• Can be used to determine whether to reject Ho.
• With the confidence interval, If 0 is present then Ho is rejected.

Example 1 Continued

• Confidence intervals illustrate this concept.
• Null hypothesis is rejected, since 0 does not fall within the CI.

P-Values in SPSS

• Most p-values are calculated assuming a two-sided hypothesis.
• The p-value is found in the Sig (2-tailed) column.
• Divide the obtained output by 2 to get the p-value for a one-sided test.