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1、Multiple Regression: Modeling of multiple possible factors for a single outcome April 16, 2014 Statistical techniques Result Endpoint Outcome Group Meas. “Cause” Group Chi-square t-test Sens./Spec. ANOVA (GLM) or Measure Logistic Correl. Log-linear Regress. “Predictor” ROC (GLM) r = -1-30-20-1001020
2、3040506000.511.522.53r = -0.87-30-20-10010203040506000.511.522.53r = -0.68-30-20-10010203040506000.511.522.53r = -0.32-30-20-10010203040506000.511.522.53r = -0.14-30-20-10010203040506000.511.522.53Null Hypothesis Possible conclusions: Null not believable, cause does predict effect. OR Null might be
3、true. Null hypothesis commonly no relationship between cause (predictor) and effect. Stat. test determines whether null hypothesis is believable. P-value reflects how often data might arise by chance if the null were true. It says nothing about chances if the null werent true. Correlations and p-val
4、ues n r p 5 0.6 0.28 50 0.6 0.000004 50 0.3 0.034 1000 0.1 0.002 Conclusion: P-value reflects sample size as much as strength of relationship Correlation (or R2 ) indicates strength of relationship. But often no test (no null hypothesis) exists to determine if one predictor is better than another. T
5、wo types of studies Comparative study: “Adjust/Control” for “other” prognostic factors in a clinical trial. Model usually uses all terms! Modeling study: Determine prognostic factors and develop predictive model. Want “simplest” model. Want to avoid over-explaining. Examples Pediatric Cardiology: ac
6、tual clinical case Time taken on daily commute to MGH: demonstrate logic Pediatric Cardiology Retrospective study of coronary lumen in children Want to control for growth. Possible independent variables: Age Weight Height BSA Acknowledgement Dr. Judith Becker Comparative Trials Wish to “Adjust/Contr
7、ol” for other factors Analysis of Covariance (ANCOVA) Correct for Imbalances Reduce “noise” improve power Correct for Imbalance Is lumen size “caused” by age YoungOldp-value Lumenmean0.1620.2070.000004 sd0.0330.033 n3127Weightmean7.4715.607.48002E-15 sd3.352.42 n3127Tx Group and LADC Lumen00.050.10.
8、150.20.250.30.511.522.5TxLADCLADC v. weight00.050.10.150.20.250.30510152025WtLADC lumenWeight and LADC Lumen by Tx00.050.10.150.20.250.30510152025WeightLADCWeight and Adjusted LADC Lumen by Tx00.050.10.150.20.250.30510152025WeightAdjusted LADCAdjusted lumen = lumen slope*(wt mean wt) Tx Group and Ad
9、justed LADC Lumen00.050.10.150.20.250.30.511.522.5TxAdjusted LADCImprove power even if balanced Group 1Group 2p n = 29n = 29 LADC0.143 mean0.1750.190 sd0.0400.039 Wt0.088 mean12.38310.129 sd4.8615.007LADC v. Treatment00.050.10.150.20.250.30.511.522.5TreatmentLADCLADC v. weight by Tx00.050.10.150.20.
10、250.30510152025WeightLADCAdjusted LADC v. weight by Tx00.050.10.150.20.250.30510152025WeightAdjusted LADCTx Group, LADC and Adjusted LADC Lumen00.050.10.150.20.250.30.511.522.5TxAdjusted LADCAdj. LADCLADCGroup 1Group 2p n = 29n = 29 LADC0.143 mean0.1750.190 sd0.0400.039 Wt0.088 mean12.38310.129 sd4.
11、8615.007 Adj. LADC0.001 mean 0.1690.196 sd0.0290.027Effect of Inclusion of non- prognostic covariates LADC: df = 56 : Effective n of 58 Adj. LADC: df = 55 : Effective n of 57 Every covariate in model effectively reduces n by 1. In a comparative study of pediatric lumen, would include all of age, wei
12、ght, height and BSA as covariates. Developing a Predictive Model Modeling Commute Endpoint: Total time on daily commute from home to MGH on various trips. All trips went Home Newton Corner Brighton Leverett Circle Parking Garage Might know some of : Distance travelled (miles) Peak Mph Gas consumed (
13、gallons) # stops for groceries, etc. # other drivers sworn at etc. for some legs. Some are correlated with time. Which one, or which combination, provides the best predictor of the time? Combining multiple predictors Might have # drivers sworn at Home Newton Corner Brighton Leverett Circle Leverett
14、Circle Garage Each positively correlated to travel time. Sum has greater R2 than any individual predictor. Combining apples and oranges Might have: gas consumed for home Newton Corner. distance traveled for Brighton Lev Circle # drivers sworn at for Lev Circle garage Each positively correlated to to
15、tal travel time. Weighted sum has greater R2 than any individual Apples v oranges The multiplier does not have any indication of the strength of the relationship. Could standardize by converting each predictor to a z-score. Get “” coeff. Testing for the effect of adding a predictor to the model Addi
16、ng predictors to a model can never decrease the R2. Removing predictors from a model can never increase the R2. The statistical tests of modeling determine whether adding an independent variable to the model significantly increased the R2. Multiple Regression Overall R2 and p-value test null hypothesis that all coefficients equal zero. Can test : R2(gas consumed leg1 & # d
