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Part One: Pearson Correlation Suppose we follow a 65-year-old...

Part One: Pearson Correlation

Suppose we follow a 65-year-old woman with low bone density for 8 years; she is treated with alendronate throughout the life of the study. Bone density is measured at various times including at baseline (time = 0 months). We are interested in the strength of the association between time from baseline (the independent variable or X, measured in months) and bone density (the dependent variable or Y, measured in gm/cm²). The data are displayed in the table below:

Visit (i)	Time from bassline (months) (x_i)	Bone density Lumbar spine (gm/cm²) (y_i)	X-Xbar	(X-Xbar)²	Y-Ybar	(Y-Ybar)²	(X-Xbar)(Y-Ybar)
1	0	0.797	-46.875000	2197.26563	-0.039250	0.001541	1.839844
2	8	0.806	-38.875000	1511.26563	-0.030250	0.000915	1.175969
3	18	0.817	-28.875000	833.76563	-0.019250	0.000371	0.555844
4	48	0.825	1.125000	1.26563	-0.011250	0.000127	-0.012656
5	64	0.837	17.125000	293.26563	0.000750	0.000001	0.012844
6	66	0.841	19.125000	365.76563	0.004750	0.000023	0.090844
7	79	0.886	32.125000	1032.01563	0.049750	0.002475	1.598219
8	92	0.881	45.125000	2036.26563	0.044750	0.002003	2.019344
Sum				8270.87500		0.007454	7.280250

1) Using the information in the table provided, calculate correlation coefficient (r) and show your work.

2) Interpret the size of r in two ways. First, give your interpretation the index value both in terms of direction (positive or negative) and strength (weak, moderate, or strong). Second, give your interpretation if the squared value of r.

Based on the above findings, answer the following questions.

3) Identify the dependent variable and the independent or explanatory variable in this study. Also, state the Null and Alternative hypotheses.

4) Report the correlation coefficient and P-value that should be used to test the Null hypothesis. Based on that information, what conclusion can you make about the Null hypothesis? (i.e. Reject or Fail to Reject the Null).

Part Two: Simple Linear Regression

Transient hypothyroxinemia, a common finding in premature infants, is not thought to have long-term consequences or to require treatment. We conduct a study to investigate whether hypothyroxinemia in premature infants is a cause of subsequent motor and cognitive abnormalities. Blood thyroxine values were obtained on routine screening in the first week of life from 536 infants who weighed 2000 g or less at birth and were born at 33 weeks gestation or earlier. The data in below Table is presented concerning the relationship between mean thyroxine level and gestational age.

(x) Gestational age	(y) Mean thyroxine level (mg/dl)
≤ 24^a	6.5
25	7.1
26	7.0
27	7.1
28	7.2
29	7.1
30	8.1
31	8.7
32	9.5
33	10.1

treated as 24 in subsequent analyses

Suppose we want to test the null hypothesis that there is not a linear relationship between gestational age (x_gest) and mean thyroxine level (y_thyro). We analyze the data using SAS's REG procedure and observe the results shown in Table 2 (see page 6). Based on those findings, answer the following questions.

1) Identify the dependent variable and the independent or explanatory variable(s) in this study. Also, state the Null and Alternative hypotheses.

2) Report the test statistic and P-value that should be used to test the Omnibus (or Overall) Null hypothesis. What is your conclusion about the Omnibus Null hypothesis?

3) Report and interpret the parameter estimate for gestational age from the SAS output.

4) Using the regression equation (Y = a + bX), calculate the predicted mean thyroxine level when gestational age is 34 weeks.

5) What is your interpretation of the R-square value?

6) Interpret these findings (3-4 sentences max). Your answer should (1) restate the findings, (2) note the generalizability of these results, and (3) state what these findings mean from clinical perspective.

Table 1: Output for Part One

Table 2: Output for Part Two

Math Statistics and Probability

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