Biostats ch. 12

comparing two means

Lots of questions in biology compare two groups:

• Drug versus control

• Before versus after (some event)

• Male vs female

(2 numerical variables)

different study designs (for numerical variables)

Two-sample design - each treatment group is composed of an independent, random sample of units

• Two independent groups

• Observations in one group are independent of those in the other

Paired design - both treatments are applied to every sampled unit

• Two related sets of observations

• Each observation is matched to a specific observation in the other set

• Will always be a paired test if one person was measured twice

  • ex., comparing fathers and sons, related, paired

• Paired designs are usually more powerful

  • Control for variation among sampling units

• But paired designs are often not possible

Paired Design

Examples:

• Patient weight before and after hospitalization

• Effects of sunscreen on one arm versus placebo on other arm

• Effects of environment on identical twins raised under different socioeconomic conditions

Two measurements from same sampling unit

Paired measurements are converted to a single measurement by taking the difference between them.

Example Problem: Paired Design

• In most species males compete with each other for access to females

• In many species, males with higher testosterone levels are more successful

• Does high testosterone have a trade-off of reduced immunocompetence?

Measured antibody production rate in 13 male red-winged blackbirds before and after a testosterone implant

Difference between paired measurements

1) Calculate SE

2) Df —> find t value

3) Calculate CI

2) Df

3) CI

Range includes 0 (no difference), but also consistent with a slight drop or a modest increase in immunocompetence following testosterone implant

Paired T-test

Used to test the null hypothesis that the mean difference of paired measurements equals a specific value

• H0: mean change was zero (Ud = 0)

• HA: mean change was not zero (Ud ≠ 0)

One paired samples are reduced to single measurement (d), the calculation of test statistic same as one-sample t-test

Assumptions of test:

• Sampling units are randomly sampled from the population

• Paired differences have a normal distribution in the population

Example Problem: Paired Design

• In most species males compete with each other for access to females

• In many species, males with higher testosterone levels are more successful

• Does high testosterone have a trade-off of reduced immunocompetence?

Measured antibody production rate in 13 male red-winged blackbirds before and after a testosterone implant

Difference between paired measurements

Solve Using Paired T-test

1) H0 and Ha

2) SE

3) Find t

4) Df

• H0: mean change in antibody production after testosterone implants was zero (Ud = 0)

• HA: mean change in antibody production after testosterone implants was not zero (Ud ≠ 0)

2)

3)

4)

so

Conclusion:

Fail to reject the null hypothesis that the mean change in antibody production after the testosterone implant is zero.

There is not sufficient evidence to conclude that the mean change in antibody production after testosterone implants is different from zero.