Question

17. Cholesterol Levels A family physician wanted to know if age and gender were factors that explained levels of serum cholesterol (in mg/dL) in her adult patients. She randomly selects two patients for each category of data and obtains the following results: Gender Age (years) 18–34 35–54 55 and older Female 180, 192 205, 226 218, 231 Male 175, 193 213, 222 203, 185 Source: National Center for Health Statistics Serum cholesterols are known to be approximately normally distributed and the population variances are equal. (a) What type of factorial design is this? How many replications are there within each cell? (b) What is the response variable? What are the two factors? (c) Determine if there is significant interaction between age and gender. (d) If there is no significant interaction, determine whether there is significant difference in the means for the three age groups. If there is no significant interaction, determine whether there is significant difference in the means for the genders. (e) Draw an interaction plot of the data to support the results of parts (c) and (d). (f) The residuals are normally distributed. Verify this. (g) If there is significant difference in the means for the three age groups, use Tukey’s test to determine which pairwise means differ using a familywise error rate of a = 0.05. If there is significant difference in the means for gender, use Tukey’s test to determine which pairwise means differ using a familywise error rate of a = 0.05.

Answer 1

	Age
Gender	18 - 34	35 -54	>55
Female	180	205	218
	192	226	231
Male	175	213	203
	193	222	185

a) This is a two factor factorial design and there are two replications in each cell.

b) The response variable is levels of serum cholesterol and the two factors are Gender and Age group.

c) ANOVA results

General Linear Model: Cholesterol levels versus Gender, Age group

Method

Factor coding

(-1, 0, +1)

Factor Information

Factor	Type	Levels	Values
Gender	Fixed	2	1, 2
Age group	Fixed	3	1, 2, 3

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Gender	1	310.1	310.1	2.51	0.164
Age group	2	2177.2	1088.6	8.81	0.016
Gender*Age group	2	628.2	314.1	2.54	0.159
Error	6	741.5	123.6
Total	11	3856.9

p-value for interaction is 0.159 which is greater than 0.05 hence we fail to reject null hypothesis and conclude that there is no significant interaction between gender and age group.

d) since p-value for age group is less than 0.05 hence the result is significant and we conclude that cholesterol levels in serum were significantly different in Age group.

p-value for gender is greater than 0.05 hence we fail to reject null hypothesis and and conclude that there is no significant difference between the gender serum cholesterol level.

e)

f)

g)

Comparisons for Cholesterol levels

Tukey Pairwise Comparisons: Age group

Grouping Information Using the Tukey Method and 95% Confidence

Age group	N	Mean	Grouping
2	4	216.50	A
3	4	209.25	A
1	4	185.00		B

Means that do not share a letter are significantly different.

Tukey Simultaneous Tests for Differences of Means

Difference of Age group Levels	Difference of Means	SE of Difference	Simultaneous 95% CI	T-Value	Adjusted P-Value
2 - 1	31.50	7.86	(7.38, 55.62)	4.01	0.017
3 - 1	24.25	7.86	(0.13, 48.37)	3.08	0.049
3 - 2	-7.25	7.86	(-31.37, 16.87)	-0.92	0.647

Individual confidence level = 97.80%

Tukey Simultaneous 95% CIs

the groups 2-1 and 3-1 are significantly different.