SAS Report about Analysis and Writing R Code
- Procreg data=class;
The SAS code above is used for regression analysis of Weight and Height. Regression analysis is an analysis that tries to establish a relationship between a response or a dependent variable from a function of regressors or explanatory variables and parameters.The parameters are measured so that measure of fit is optimized, the equation used in this analysis is as follows:
Where Yi is the response variable, Xi is the regress variable, βo and β1 are unknown parameters to be estimated, and εi is the error term associated with the ith observation (Crawley 34-41). In this case, we are relating a child’s weight toa child’s height. In such a scenario, a simple linear regression model with response variable weight and the regressor variable height can be written as follows:
Wghti= βo+β1Heighti +εi
The output of the above code is the default tabular output of the REG procedure for this model. Number of observations read=19, number of observation used is also 19: since there is no unobservable or missing value from the provided data.
Based on the parameter estimates by least-square estimates, the fitted regression line relationship between weight and height is described by the resulting equation
predicted Wght=-143.02692 + 3.89903 * Height.
At the least square solution, the residual sum of squares is:
SSE=∑i=119(Wghti – βo – β1 Height)2
The above is minimized and achieved value is displayed in the analysis of the variance table as the error sum of squares: 2142.48772
The fit plot below shows positive slope of fitted line. The average weight of a child varies by 3.89903 units for every unit change in height. The 1-α=95% confidence limits in the fit plot are point wise limits that cover the mean weight for a particular height with probability 0.95. The prediction limits, which are wider than the confidence limits, show the point wise limits that cover a new observation for a given height with probability 0.095. The statistic for the overall model is highly significant (=57.076, <0.0001), indicating that the model explains a significant portion of the variation in the data.
The degrees of freedom can be used in checking the accuracy of the data and model. The model degrees of freedom are one less than the number of parameters to be estimated (Dalgaard 11-24). This model estimates two parameters, βo and β1; thus, the degrees of freedom should be . The corrected total degrees of freedom are always one less than the total number of observations in the data set, in this case.
Several simple statistics follow the ANOVA table. The Root MSE is an estimate of the standard deviation of the error term. The coefficient of variation, or Coeff Var, is a unitless expression of the variation in the data. The R-square and Adj R-square are two statistics used in assessing the fit of the model; values close to 1 indicate a better fit. The R-square of 0.77 indicates that Height accounts for 77% of the variation in Weight
The function attach(DATA) attaches the data frame DATA so that it can be accessed in R. The argument lm is a model formula in which the tilde symbol (~) should be read as “described by.”
- Estimated intercept α
- Estimated slope β
The best-fitting is a straight line seen to be;
Wght= slope(α) + Height(β)
The result of lm is a model object. To obtain more information on the linear model (lm), we proceed to write the following function in R,
This gives a superficial view of the distribution of the residuals that may be used as a quick check of the distribution assumptions. The average of the residuals is zero by definition, so the median should not be far from zero, and the minimum and the maximum should roughly be equal in absolute value.
The regression coefficient and the intercept are accompanied by standard errors, t-test and the p-values (probability values). The symbols to the righty are graphical indicators of the level of significance.
lm(formula = Wght ~ Height)
Min 1Q Median 3Q Max
-17.6807 -6.0642 0.5115 9.2846 18.3698
Estimate Std. Error t value Pr(>|t|)
(Intercept) -143.0269 32.2746 -4.432 0.000366 ***
Height 3.8990 0.5161 7.555 7.89e-07 ***
Signif.codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 11.23 on 17 degrees of freedom
Multiple R-squared: 0.7705, Adjusted R-squared: 0.757
F-statistic: 57.08 on 1 and 17 DF, p-value: 7.887e-07
1 2 3 4 5 6
-13.5061701 6.7317083 -13.5797581 0.6678176 -2.0615036 2.6124840
7 8 9 10 11 12
-5.6350916 11.8375266 -16.6624734 12.4841326 -6.4933343 -17.6807278
13 14 15 16 17 18
0.5115143 -4.2585944 12.2967391 18.3697570 14.7918904 3.8326780
>plot (Wght, Height, col=”red”)
1 2 3 4 5 6 7 8
126.00617 77.26829 111.57976 101.83218 104.56150 80.38752 90.13509 100.66247
9 10 11 12 13 14 15 16
100.66247 87.01587 56.99333 107.68073 76.48849 116.25859 137.70326 109.63024
17 18 19
118.20811 81.16732 116.25859
> predicted=predict (lm.Wght)
>plot (residuals, predicted)
Crawley, Michael J. Statistics: An Introduction Using R. Hoboken: John Wiley & Sons, 2011. Print.
Dalgaard, Peter. Introductory Statistics with R. New York: Springer, 2008. Print.
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