Lesson 27: Correlation and Simple RegressionLesson 27: Correlation and Simple Regression
In this lesson, we investigate statistical analyses that are typically performed when dealing with two or more continuous numeric variables. Specifically, we investigate:
- the GPLOT procedure, to create publication quality x-y scatter plots of any two numeric variables in a SAS data set
- the CORR procedure, to compute various correlation coefficients between two or more numeric variables in a SAS data set
- the REG procedure, to perform a regression analysis on any subset of numeric variables in a SAS data set
Upon completing this lesson, you should be able to do the following:
- use the CORR procedure to tell SAS to calculate Pearson correlation coefficients among a set of numeric variables
- use the CORR procedure's SPEARMAN, KENDALL, and HOEFFDING options to tell SAS to calculate alternative coefficients
- read typical correlation procedure output in order to be able to extract the calculated correlations and their associated P-values
- use the CORR procedure's WITH statement to tell SAS to calculate only the correlation coefficients among the variables in the WITH and VAR statements
- understand how sample size can affect the significance of a correlation coefficient
- interpret a correlation coefficient
- use the CORR procedure's PARTIAL statement to tell SAS to calculate partial correlations among variables
- use the CORR procedure's BEST = n option to tell SAS to print only the first n of the ordered estimated correlations
- use the REG procedure to compute a regression equation between two numeric variables
- use the REG procedure's MODEL statement to tell SAS which variable to treat as the response variable and which variable to treats as the predictor variables
- read the typical SAS output from a regression analysis to pull off key information, such as parameter estimates, confidence intervals, and P-values
- use the REG procedure's PLOT statement to request residual diagnostic plots
- use the GPLOT procedure to request plots containing estimated regression equations, 95% confidence intervals about the mean of y, and 95% prediction intervals about the individual y-values
- use the REG procedure to conduct a regression analysis involving quadratic terms
- use the REG procedure to conduct a regression analysis involving transformed variables
Chapter 5 of the textbook.
27.1 - Lesson Notes27.1 - Lesson Notes
C. Significance of a Correlation Coefficient
Page 163. The point the authors make in the first paragraph is a very important one. If you have a large data set, it is quite possible that you'll have a small enough P-value to conclude that the population correlation is significantly different from 0, when in fact it is not all that much different from 0. You'll want to make sure you use common sense when interpreting the results.
The point the authors make in the second paragraph is equally important. Always remember the mantra: correlation does not imply causation.
Page 164. The authors present a great example. I just wished they had taken it a little bit further. What they are getting at is the possibility of getting a sample correlation coefficient that leads to rejecting the null hypothesis of a zero population correlation coefficient, when in fact the population correlation coefficient is zero. You might recall from your statistical studies that we call this a Type I error.
If you look at the data in the plot, it doesn't look like there is much of a correlation between the x and y variables. If you think of all of the dots on the plot as the population, and the black dots on the plot as the sample, then you can get the idea that by chance you might get a sample that leads to concluding the population correlation differs from 0 when in fact it doesn't. Since the authors didn't provide the data, I eyeballed the black dots on the plot and ran the CORR procedure on the data:
OPTIONS PS = 58 LS = 72 NODATE NONUMBER; DATA HOSP_PATIENTS; INPUT #1 @1 ID $3. @4 DATE1 MMDDYY8. @12 HR1 3. @15 SBP1 3. @18 DBP1 3. @21 DX1 3. @24 DOCFEE1 4. @28 LABFEE1 4. #2 @4 DATE2 MMDDYY8. @12 HR2 3. @15 SBP2 3. @18 DBP2 3. @21 DX2 3. @24 DOCFEE2 4. @28 LABFEE2 4. #3 @4 DATE3 MMDDYY8. @12 HR3 3. @15 SBP3 3. @18 DBP3 3. @21 DX3 3. @24 DOCFEE3 4. @28 LABFEE3 4. #4 @4 DATE4 MMDDYY8. @12 HR4 3. @15 SBP4 3. @18 DBP4 3. @21 DX4 3. @24 DOCFEE4 4. @28 LABFEE4 4.; FORMAT DATE1-DATE4 MMDDYY10.; DATALINES; 0071021198307012008001400400150 0071201198307213009002000500200 007 007 0090903198306611007013700300000 009 009 009 0050705198307414008201300900000 0050115198208018009601402001500 0050618198207017008401400800400 0050703198306414008401400800200 ; RUN; PROC PRINT data = HOSP_PATIENTS; RUN;
This is the portion of the output that makes the point:
Pearson Correlation Coefficient, N=10
Prob > |r| under H0: Rho=0
As suspected, the sample correlation is large (0.91252) leading to a small P-value (0.0002). Therefore, we would reject the null hypothesis of a zero population correlation coefficient, when in fact the population correlation coefficient is zero. We would indeed be committing a Type I error.
F. Linear Regression
Page 168. In general, the 95% confidence interval for the slope is obtained by taking the parameter estimate of the slope (11.19127) and adding and subtracting 2 standard errors (2 × 1.2178). The calculation is as follows:
11.19127 - (2 × 1.2178) = 11.19127 - 2.4356 = 8.75 11.19127 + (2 × 1.2178) = 11.19127 + 2.4356 = 13.63
In the case when the sample size is small, as it is here, we replace the 2 with the appropriate t-value. As the authors, explain the appropiate t-value here is 2.57. Then, the calculation is:
11.19127 - (2.57 × 1.2178) = 11.19127 - 3.1297 = 8.06 11.19127 + (2.57 × 1.2178) = 11.19127 + 3.1297 = 14.32
Page 169. Here, since the reported R-square value is 0.9441, we can say that 94.4% of the variation in weight can be explained by height.
H. Producing a Scatter Plot and the Regression Line
Page 170. The authors use the graphic version of the scatter plot procedure. You can create a character-based plot as well using the PLOT procedure. In that case, the code would be the same except PLOT would replace GPLOT:
PROC PLOT data = corr_eg; PLOT WEIGHT*HEIGHT; RUN;
If you haven't worked with the GPLOT procedure before, you should be aware that SAS displays the results in a new graph window. You might want to run the code so that you can see this for yourself:
DATA CORR_EG; INPUT GENDER $ HEIGHT WEIGHT AGE; DATALINES; M 68 155 23 F 61 99 20 F 63 115 21 M 70 205 45 M 69 170 . F 65 125 30 M 72 220 48 ; RUN; SYMBOL VALUE = DOT COLOR = BLACK; PROC GPLOT DATA=CORR_EG; title 'Plot of Weight vs. Height'; PLOT WEIGHT*HEIGHT; RUN; QUIT;
By the way, you'll notice that this program ends with a QUIT; statement. If you leave it out, the GPLOT procedure will continue to run in the background. You can see this for yourself by removing the QUIT; statement, and re-running the program. Then, look at the blue bar at the very top of your SAS window. You should see:
In general, if you want a quick-and-dirty plot, the PLOT procedure will probably suffice. On the other hand, if you want a publication quality plot, you'll want to use the GPLOT procedure.
Page 173. The end of the first sentence on this page should say "... interval about the individual y-values (RLCLI95)."
J. Transforming Data
Page 177. A common programming mistake is to try to transform the variables right in the model statement:
MODEL hr = log(dose);
Don't worry... SAS will squawk at you to let you know that you've made a mistake. If it does, just remind yourself that you need to make the transformation in the DATA step, not in the MODEL statement.
27.2 - Summary27.2 - Summary
In this lesson, we investigated correlation and regression analyses that are typically performed when dealing with two or more continuous numeric variables.
The homework for this lesson will give you more practice with conducting these analyses in SAS. Then, you can use the methods to perform correlation and regression analyses on your own data!