14. Do left-handed people live shorter lives than right-handed people? A study o

Question

14. Do left-handed people live shorter lives than right-handed people? A study of this question examined a sample of 949 death records and contacted next of kin to determine handedness. Note that there are many possible definitions of "left-handed." The researchers examined the effects of different definitions on the results of their analysis and found that their conclusions were not sensitive to the exact definition used. For the results presented here, people were defined to be right-handed if they wrote, drew, and threw a ball with the right hand. All others were defined to be left-handed. People were classified by sex (female or male) and handedness (left or right), and a 2 2 ANOVA was run with the age at death as the response variable. The F statistics were 22.36 (handedness), 37.44 (sex), and 2.10 (interaction). The following marginal mean ages at death (in years) were reported: 77.39 (females),71.32 (males), 75.00 (right-handed), and 66.03 (left-handed).

(a)

For each of the F statistics given, find the degrees of freedom and an approximate P-value. Summarize the results of these tests.

(b)

Using the information given, write a short summary of the results of the study.

Explanation / Answer

(A)

We have a 2 * 2 ANOVA

There is a row effect (effect due to sex)

No. of rows = No of groups of sex = 2 (male and female)

=> r = 2

There is a column effect (effcet due to left handedness and right handedness)

No. of columns = No. of groups due to handedness = 2 (left and right)

=> r = 2

There is an interaction effect due to sex and handedness.

No. of samples = 949

=> n = 949

We break total sum of squares as follows:

Total sum of squares (df = n - 1 = 948)

=

Sum of squares due to sex effect (df = r - 1 = 1)

+ Sum of squares due to handedness effect (df = c - 1 = 1)

+ Interaction effect (df = (r - 1) * (c - 1) = 1)

+ Error sum of squares (df = 948 - 1 - 1 - 1 = 945)

So, degrees of freedom for the F-statistics are:

sex -> (1, 945)

handedness -> (1, 945)

interaction -> (1, 945)

Approximate p - values:

p-value for F statistic for handedness:

P(F(1, 945) > 22.36) = 2.604464e-06

p-value for F statistic for sex:

P(F(1, 945) > 37.44) = 1.37921e-09

p-value for F statistic for interaction:

P(F(1, 945) > 2.10) = 0.1476309

These values are calculated in R as follows:

> 1 - pf(22.36, 1, 945)
[1] 2.604464e-06

> 1 - pf(37.44, 1, 945)
[1] 1.37921e-09

> 1 - pf(2.1, 1, 945)
[1] 0.1476309

The p-values of the F statistics for sex and handedness are statistically significant but the p-value of the F statistics for the interaction effect is not statistically significant.

(b)

Summary:

The p-value for the F-statistic for the effect due to sex is statistically significant.

The mean ages at death are 77.39 for females and 71.32 for females.

So, the mean ages at death is significantly higher for females than males.

The p-value for the F-statistic for the effect due to handedness is statistically significant.

The mean ages at death are 75.00 for right-handed and 66.03 for left-handed.

So, the mean ages at death is significantly higher for right-handed people than left handed people.

However the p-value for the effect due to interacton between sex and handedness is not statistically significant.

Hence, there is no significant effect difference in mean ages at death due to an intercation effect between sex and handedness.

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