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Creating Graphs on Saturn GOPTIONS DEVICE = png HTITLE = 2 HTEXT = 1.5 GSFMODE = replace;

Creating Graphs on Saturn GOPTIONS DEVICE = png HTITLE = 2 HTEXT = 1.5 GSFMODE = replace; PROC REG DATA =agebp; MODEL sbp = age; PLOT sbp*age; RUN ; This will create file sasgraph.png Transfer file to PC (binary mode) 2. Open Word 3. Choose Insert picture from file

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Creating Graphs on Saturn GOPTIONS DEVICE = png HTITLE = 2 HTEXT = 1.5 GSFMODE = replace;

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  1. Creating Graphs on Saturn • GOPTIONSDEVICE = png HTITLE=2HTEXT=1.5 • GSFMODE = replace; • PROCREGDATA=agebp; • MODEL sbp = age; • PLOT sbp*age; • RUN; • This will create file sasgraph.png • Transfer file to PC (binary mode) • 2. Open Word • 3. Choose Insert picture from file • PROCREGDATA=agebp LP; • MODEL sbp = age; • PLOT sbp*age; • RUN;

  2. Multiple Linear Regression • More than 1 independent variable • See how combinations of several variables are associated with and can predict the dependent variable. How much of the total variability can be explained? • Control for confounding (interested in the effect of one variable but want to “adjust” for another variable) • Explore interactions PROCREG DATA=datasetname; MODELdepvar = x1; MODELdepvar = x1 x2; MODELdepvar = x1 x2 x3; RUN;

  3. Question Explored Using Multiple Regression • How much of the variation in test scores among school districts can be explained by several district characteristics? • Is calcium intake related to BP independent of age? • Is the relationship between age and BP the same for men and women.

  4. Reminder • Y variable is continuous and is normally distributed for each combination of X’s with the same variability • X variables can be continuous or indicator variables and do not need to be normally distributed

  5. 2 Factors • Y = b0 + b1X1 • Y = b0 + b2X2 • Y = b0 + b1X1 + b2X2 • Do you get the same slope in models 1 and 3

  6. Control for confounding Both SLR models for each cohort significant Overall not significant (negative confounding)

  7. Multiple Regression Equation • The equation that describes how the mean value of y is related to x1, x2, . . . xp . my = 0 + 1x1 + 2x2 + . . . + pxp b0=Mean of y when all x variables are equal to 0 bi = change in mean y corresponding to a 1 unit change in xi considering all other predictors fixed Implied: The impact of x1 is the same for each of the other values of x2, x3, … xp

  8. Multiple Regression Model • The equation that describes how the dependent variable y is related to the independent variables x1, x2, . . . xp and an error term is called the multipleregression model. y = b0 + b1x1 + b2x2 +. . . + bpxp + e ereflects how individuals deviate from others with the same values of x’s

  9. Estimated Multiple Regression Equation • The estimated multiple regression equation is: y = b0 + b1x1 + b2x2 + . . . + bpxp ^ bi estimates bi yis estimated (or predicted) value for a set of x’s ^

  10. Estimation • Least Squares Criterion • Computation of Coefficients Values The formulas for the regression coefficients b0, b1, b2, . . . bp involve the use of matrix algebra. We will use SAS to perform the calculations. ^

  11. Find the best multidimensional plane

  12. Testing for Significance: Global Test • Hypotheses H0: 1 = 2 = . . . = p = 0 Ha: One or more of the parameters is not equal to zero. • Test Statistic F = MSR/MSE • Rejection Rule Reject H0 if F > F where F is based on an F distribution with p d.f. in the numerator and n - p - 1 d.f. in the denominator.

  13. Testing for Significance: Individualb’s • Hypotheses H0: i = 0 Ha: i = 0 • Test Statistic • Rejection Rule Reject H0 for small or large t Meaning: Is Xi related to Y after taking into account all other variables in the model

  14. Possibilities • X1 is related to Y alone but after adjusting for X2, then X1 is no longer related to Y • X1 is not related to Y alone but after adjusting for X2, then X1 is related to Y • Relation of X1 with Y1 gets stronger after adjusting for X2 • Relation of X1 with Y gets weaker after adjusting for X2

  15. Pulmonary Function Example • Dependent Variable: Forced Expired Volume (FEV1.0) • Independent Variables: • Age of person • Smoking status of person • Questions: • Is age related to FEV independent of smoking status • Is smoking status related to FEV independent of age • How much of the variability in FEV is explained by age and smoking combined

  16. Model for FEV Example Y = b0 + b1X1 + b2X2 X1 = smoking status (1=smoker, 0=nonsmoker) X2 = age Smokers FEV = b0 + b1 + b2age Non Smokers FEV = b0 + b2age

  17. Interpretation of Parameters Smokers FEV = b0 + b1 + b2age Non Smokers FEV = b0 + b2age b1 is the effect of smoking for fixed levels of age b2 is the effect of age pooled over smokers and non-smokers This model assumes the relation of age to FEV is the same for smokers and non-smokers

  18. DATA fev; INFILE DATALINES; INPUT age smk fev; DATALINES; 28 1 4.0 30 1 3.9 30 1 3.7 31 1 3.6 54 0 2.9 More data

  19. PROCMEANS; VAR fev; CLASS smk; RUN; The MEANS Procedure Analysis Variable : fev N smk Obs N Mean Std Dev Minimum Maximum 0 15 15 3.6000000 0.4208834 2.9000000 4.3000000 1 15 15 3.2933333 0.5257195 2.2000000 4.000000

  20. PROCCORR DATA=fev; Pearson Correlation Coefficients, N = 30 Prob > |r| under H0: Rho=0 age smk fev age 1.00000 -0.12788 -0.73024 0.5007 <.0001 smk -0.12788 1.00000 -0.31620 0.5007 0.0887 fev -0.73024 -0.31620 1.00000 <.0001 0.0887

  21. PROCREG; MODEL fev = age smk ; RUN; Dependent Variable: fev Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 SSR 4.96510 2.48255 32.08 <.0001 Error 27 SSE 2.08957 0.07739 Corrected Total 29 SST 7.05467 Root MSE 0.27819 R-Square 0.7038 Dependent Mean 3.44667 Coeff Var 8.07136 Tests Ho: b1 = 0; b2 =0 Proportion of variance explained by both variables

  22. PROCREG; MODEL fev = age smk ; MODEL fev = age ; MODEL fev = smk ; RUN; Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 5.58114 0.27653 20.18 <.0001 age 1 -0.04702 0.00634 -7.42 <.0001 smk 1 -0.40384 0.10242 -3.94 0.0005 Intercept 1 5.24787 0.32456 16.17 <.0001 age 1 -0.04382 0.00775 -5.66 <.0001 Intercept 1 3.60000 0.12295 29.28 <.0001 smk 1 -0.30667 0.17388 -1.76 0.0887 R2 = .7038 R2 = .5333 R2 = .1000

  23. PROCREG; MODEL fev = age smk; PROCREG; MODEL fev = age ; WHERE smk = 0; PROCREG; MODEL fev = age ; WHERE smk = 1; Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 5.58114 0.27653 20.18 <.0001 age 1 -0.04702 0.00634 -7.42 <.0001 smk 1 -0.40384 0.10242 -3.94 0.0005 Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 5.24764 0.38050 13.79 <.0001 age 1 -0.03911 0.00887 -4.41 0.0007 Intercept 1 5.50002 0.36163 15.21 <.0001 age 1 -0.05508 0.00885 -6.22 <.0001 Non-smokers Smokers

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