BIOSTATISTICS 755 HOMEWORK 1 -Solved

This question will use data on Orthodontic Measurements on Children. The data are from a 
study of dental growth measurements of the distance (mm) from the center of the pituitary 
gland to the pteryomaxillary fissure. Measurements were obtained on 11 girls and 16 boys 
at ages 8, 10, 12, and 14. Scientific goals of the study are to: 
• determine whether distances over time are larger for boys than for girls, and 
• whether the rate of change of distance over time is similar for boys and girls. 
The data is available on the website in the file dental.txt, which contains the data and 
information about the study and variables. 
Please use this dataset to answer the following questions. Note that dental.txt is given in 
the wide format. This format is good for questions 1, 4 and 5. For question 2 the long format 
will be needed. Please see the example from class on how to go from the wide to the long 
format. 
1. ( Read the data into SAS and calculate sample means, standard deviations 
and variances of the distance measurements at each occasion. 
Variable 
Mean Std Dev Variance 
Age8 
Age10 
Age12 
Age14 
22.1851852 
23.1666667 
24.6481481 
26.0925926 
2.4343225 
2.1572775 
2.8175781 
2.7666873 
5.9259259 
4.6538462 
7.9387464 
7.6545584 
2. Construct three time plots for data. For these plots: 
(a)  Make all the colors differ by subject. 
Proc SGplot data = dent_long; 
series x=age y=response / group =ID LineAttrs= (pattern=1); 
run;(b) (10 points) Make a panel plot which separate panels for boys and girls (make all lines 
black). 
Proc SGpanel data = dent_long; 
PanelBy gender / columns=2; 
series x=age y=response / group =ID LineAttrs= (pattern=1 
color="black"); 
run;(c) (10 points) Repeat this plot adding a blue mean line to each panel. 
proc sort data=dent_long; 
by age gender; 
run; 
proc means mean data=dent_long; 
by age gender; 
var response; 
output out=dent_long_mn mean=age_response_mn; 
run; 
data stacked_dent2; 
set dent_long dent_long_mn; 
run; 
proc sort data=stacked_dent2; 
by gender; 
run; 
proc sgpanel data= stacked_dent2; 
panelby gender / columns=2; 
series x=age y=response / group= ID lineattrs=(pattern=1 
color='black'); 
series x=age y=age_response_mn / lineattrs=(pattern=1 
color="blue");run; 

3. Using the plot in 2c: 
(a) (discuss the differences in the means for boys and girls, and 
The means for both boys and girls increase over time. The boys appear to have a slightly 
higher starting point, and a slope that is larger. This results in what appears to be a bigger 
difference in the means at age 14 versus age 8. 
(b) (10 points) discuss the pattern in variation over time. 
For both boys and girls, I would say the spread of the data is relatively consistent 
overtime and similar to each other. The within subject variation for the boys is markedly 
larger than the within subject variation for the girls. There appears to be an increase in 
measurement error for the boys versus girls. 

4. (1Calculate the 4 x 4 covariance and correlation matrices for the four 
repeated measures. Does correlation appear to vary by the time difference between 
measurements? Covariance Matrix, DF = 26 
Age8 
Age10 
Age12 
Age14 
Age8 5.925925926 3.285256410 4.875356125 4.039886040 
Age10 3.285256410 4.653846154 3.858974359 4.532051282 
Age12 4.875356125 3.858974359 7.938746439 6.197293447 
Age14 4.039886040 4.532051282 6.197293447 7.654558405 
Simple Statistics 
Variable N 
Mean Std Dev 
Sum Minimum Maximum 
Age8 
27 22.18519 2.43432 599.00000 
16.50000 
27.50000 
Age10 
27 23.16667 2.15728 625.50000 
19.00000 
28.00000 
Age12 
27 24.64815 2.81758 665.50000 
19.00000 
31.00000 
Age14 
27 26.09259 2.76669 704.50000 
19.50000 
31.50000 
Pearson Correlation Coefficients, N = 27 
Prob > |r| under H0: Rho=0 
Age8 
Age10 
Age12 
Age14 
Age8 
1.00000 0.62558 
0.0005 
0.71081 
<.0001 
0.59983 
0.0009 
Age10 
0.62558 
0.0005 
1.00000 0.63488 
0.0004 
0.75933 
<.0001 
Age12 
0.71081 
<.0001 
0.63488 
0.0004 
1.00000 0.79500 
<.0001 
Age14 
0.59983 
0.0009 
0.75933 
<.0001 
0.79500 
<.0001 
1.00000 
Covariance Matrix, DF = 10 
Age8 
Age10 
Age12 
Age14 
Age8 4.513636364 3.354545455 4.331818182 4.356818182 
Age10 3.354545455 3.618181818 4.027272727 4.077272727 
Age12 4.331818182 4.027272727 5.590909091 5.465909091 
Age14 4.356818182 4.077272727 5.465909091 5.940909091Overall, the correlation appears to be relatively consistent over time. 
5. (10 points) Calculate the 4 x 4 covariance and correlation matrices for the four 
repeated measures separately for boys and girls. Comment on the differences in the 
variance and correlation between boys and girls. 
For females: 
Simple Statistics 
Variable N 
Mean Std Dev 
Sum Minimum Maximum 
Age8 
11 21.18182 2.12453 233.00000 
16.50000 
24.50000 
Age10 
11 22.22727 1.90215 244.50000 
19.00000 
25.00000 
Age12 
11 23.09091 2.36451 254.00000 
19.00000 
28.00000 
Age14 
11 24.09091 2.43740 265.00000 
19.50000 
28.00000 
Pearson Correlation Coefficients, N = 11 
Prob > |r| under H0: Rho=0 
Age8 
Age10 
Age12 
Age14 
Age8 
1.00000 0.83009 
0.0016 
0.86231 
0.0006 
0.84136 
0.0012 
Age10 
0.83009 
0.0016 
1.00000 0.89542 
0.0002 
0.87942 
0.0004 
Age12 
0.86231 
0.0006 
0.89542 
0.0002 
1.00000 0.94841 
<.0001 
Age14 
0.84136 
0.0012 
0.87942 
0.0004 
0.94841 
<.0001 
1.00000 
For Males: 
Covariance Matrix, DF = 15 
Age8 
Age10 
Age12 
Age14 
Age8 6.016666667 2.291666667 3.629166667 1.612500000 
Age10 2.291666667 4.562500000 2.193750000 2.810416667 
Age12 3.629166667 2.193750000 7.032291667 3.240625000 
Age14 1.612500000 2.810416667 3.240625000 4.348958333Simple Statistics 
Variable N 
Mean Std Dev 
Sum Minimum Maximum 
Age8 
16 22.87500 2.45289 366.00000 
17.00000 
27.50000 
Age10 
16 23.81250 2.13600 381.00000 
20.50000 
28.00000 
Age12 
16 25.71875 2.65185 411.50000 
22.50000 
31.00000 
Age14 
16 27.46875 2.08542 439.50000 
25.00000 
31.50000 
Pearson Correlation Coefficients, N = 16 
Prob > |r| under H0: Rho=0 
Age8 
Age10 
Age12 
Age14 
Age8 
1.00000 0.43739 
0.0902 
0.55793 
0.0247 
0.31523 
0.2343 
Age10 
0.43739 
0.0902 
1.00000 0.38729 
0.1383 
0.63092 
0.0088 
Age12 
0.55793 
0.0247 
0.38729 
0.1383 
1.00000 0.58599 
0.0171 
Age14 
0.31523 
0.2343 
0.63092 
0.0088 
0.58599 
0.0171 
1.00000 
Overall, the within subject correlation for the girls is higher than it is for the boys. The 
correlation is also slightly more consistent for the girls. The variation is higher for girls than it is 
for boys. 

6. ( Recall the 4 correlations “truths” from the second set of slides. Describe on 
what these truths mean in the context of this example. In your description, try to be 
more specific than restating each “truth” in context. For example, don’t use the word 
‘correlation’ in your description, think about what the “truth” tells you about the data if 
taken as fact. 
The four correlation truths in this context are: 
1. the correlations are positive: 
Measurements from the same child are likely to be more alike than measurements 
from different children. That is, if a child’s 1st measurement is higher than average, we 
expect their remaining measurements to be higher than average. 
2. the correlations often decrease with increasing time separation 
If a child’s 1st measurement is higher than average, we are more confident in their 
2nd measurement being higher than average than their 3rd or 4th measurement being higher 
than average.3. the correlations between repeated measures rarely ever approach zero 
If a child’s 1st measurement is higher than average, than a later measurement is 
more likely to be higher than average than lower than average no matter how long between 
the measurements. 
4. the correlation between a pair of repeated measures taken very closely rarely approaches 
one. 
No matter how close in time two measurements are taken, there will be some 
difference between them. 
7. (State three potential sources of variability in the data in context of the 
problem. Using the time plot you constructed in 2(c) give your hypothesized ordering of 
which source of variability is the largest. 
Three sources of variability: 
1. differences between the kids 
2. day to day or month to month differences in the true dental differences, 
3. error when measuring the distances. 
For girls, I think that for 1 is the largest, followed by 3, then 2. 
For the boys, it’s a little harder to tell. It might be the same as the girls, but it might be 
that 3 is the largest, followed by 1, then 2.