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EPI 809/Spring 2008 1 Probability Distribution of Random Error.

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Presentation on theme: "EPI 809/Spring 2008 1 Probability Distribution of Random Error."— Presentation transcript:

1 EPI 809/Spring Probability Distribution of Random Error

2 EPI 809/Spring Regression Modeling Steps  1.Hypothesize Deterministic Component  2.Estimate Unknown Model Parameters  3.Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error Estimate Standard Deviation of Error  4.Evaluate Model  5.Use Model for Prediction & Estimation

3 EPI 809/Spring Linear Regression Assumptions Assumptions of errors    n Assumptions of errors    n - Gauss-Markov condition - Gauss-Markov condition 1. Independent errors 2. Mean of probability distribution of errors is 0 3. Errors have constant variance σ 2, for which an estimator is S 2 4. Probability distribution of error is normal 5. Potential violation of G-M condition.

4 EPI 809/Spring Error Probability Distribution

5 EPI 809/Spring Random Error Variation

6 EPI 809/Spring Random Error Variation  1.Variation of Actual Y from Predicted Y

7 EPI 809/Spring Random Error Variation  1.Variation of Actual Y from Predicted Y  2.Measured by Standard Error of Regression Model Sample Standard Deviation of , s Sample Standard Deviation of , s ^

8 EPI 809/Spring Random Error Variation  1.Variation of Actual Y from Predicted Y  2.Measured by Standard Error of Regression Model Sample Standard Deviation of , s Sample Standard Deviation of , s  3. Affects Several Factors Parameter Significance Parameter Significance Prediction Accuracy Prediction Accuracy ^

9 EPI 809/Spring Evaluating the Model Testing for Significance

10 EPI 809/Spring Regression Modeling Steps  1. Hypothesize Deterministic Component  2.Estimate Unknown Model Parameters  3.Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error Estimate Standard Deviation of Error  4.Evaluate Model  5.Use Model for Prediction & Estimation

11 EPI 809/Spring Test of Slope Coefficient  1. Shows If There Is a Linear Relationship Between X & Y  2.Involves Population Slope  1  3.Hypotheses H 0 :  1 = 0 (No Linear Relationship) H 0 :  1 = 0 (No Linear Relationship) H a :  1  0 (Linear Relationship) H a :  1  0 (Linear Relationship)  4.Theoretical basis of the test statistic is the sampling distribution of slope

12 EPI 809/Spring Sampling Distribution of Sample Slopes

13 EPI 809/Spring Sampling Distribution of Sample Slopes

14 EPI 809/Spring Sampling Distribution of Sample Slopes  All Possible Sample Slopes  Sampl e 1:2.5  Sampl e 2:1.6  Sampl e 3:1.8  Sampl e 4:2.1 : : Very large number of sample slopes

15 EPI 809/Spring Sampling Distribution of Sample Slopes  All Possible Sample Slopes  Samp le 1:2.5  Samp le 2:1.6  Samp le 3:1.8  Samp le 4:2.1 : : large number of sample slopes Sampling Distribution 1111 1111 S ^ ^

16 EPI 809/Spring Slope Coefficient Test Statistic

17 EPI 809/Spring Test of Slope Coefficient Rejection Rule  Reject H 0 in favor of H a if t falls in colored area  Reject H 0 for H a if P-value = P(T>|t|) |t|) < α T=t (n-2) 0 t 1-α/2, (n-2) Reject H 0 0 α/2 -t 1-α/2, (n-2) α/2

18 EPI 809/Spring Test of Slope Coefficient Example  Reconsider the Obstetrics example with the following data: Estriol (mg/24h) B.w. (g/1000)  Is the Linear Relationship between Estriol & Birthweight significant at.05 level?

19 EPI 809/Spring Solution Table For β’s

20 EPI 809/Spring Solution Table for SSE Birth weight =y Estriol =x Predicted =y=β 0 + β 1 x (Obs-pred) 2 =( y - y) SSE=1.1 ^^^^

21 EPI 809/Spring Test of Slope Parameter Solution  H 0 :  1 = 0  H a :  1  0   .05  df  = 3  Critical Value(s): Test Statistic:

22 EPI 809/Spring Test Statistic Solution From Table

23 EPI 809/Spring Test of Slope Parameter  H 0 :  1 = 0  H a :  1  0   .05  df  = 3  Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence of a linear relationship

24 EPI 809/Spring Test of Slope Parameter Computer Output  Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept Estriol  t =  k / S  P-Value SS kk k k ^ ^ ^ ^

25 EPI 809/Spring Measures of Variation in Regression  1.Total Sum of Squares (SS yy ) Measures Variation of Observed Y i Around the Mean  Y Measures Variation of Observed Y i Around the Mean  Y  2.Explained Variation (SSR) Variation Due to Relationship Between X & Y Variation Due to Relationship Between X & Y  3.Unexplained Variation (SSE) Variation Due to Other Factors Variation Due to Other Factors

26 EPI 809/Spring Variation Measures Total sum of squares (Y i -  Y) 2 Unexplained sum of squares (Y i -  Y i ) 2 ^ Explained sum of squares (Y i -  Y) 2 ^ YiYiYiYi

27 EPI 809/Spring  1.Proportion of Variation ‘Explained’ by Relationship Between X & Y Coefficient of Determination 0  r 2  1

28 EPI 809/Spring Coefficient of Determination Examples r 2 = 1 r 2 =.8r 2 = 0

29 EPI 809/Spring Coefficient of Determination Example  Reconsider the Obstetrics example. Interpret a coefficient of Determination of  Answer: About 82% of the total variation of birthweight Is explained by the mother’s Estriol level.

30 EPI 809/Spring r 2 Computer Output Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var r 2 adjusted for number of explanatory variables & sample size S r2r2

31 EPI 809/Spring Using the Model for Prediction & Estimation

32 EPI 809/Spring Regression Modeling Steps  1.Hypothesize Deterministic Component  2.Estimate Unknown Model Parameters  3.Specify Probability Distribution of Random Error Term-Estimate Standard Deviation of Error  4.Evaluate Model  5.Use Model for Prediction & Estimation

33 EPI 809/Spring Prediction With Regression Models What Is Predicted? Population Mean Response E(Y) for Given X Population Mean Response E(Y) for Given X Point on Population Regression LinePoint on Population Regression Line Individual Response (Y i ) for Given X Individual Response (Y i ) for Given X

34 EPI 809/Spring What Is Predicted?

35 EPI 809/Spring Confidence Interval Estimate of Mean Y

36 EPI 809/Spring Factors Affecting Interval Width  1.Level of Confidence (1 -  ) Width Increases as Confidence Increases Width Increases as Confidence Increases  2.Data Dispersion (s) Width Increases as Variation Increases Width Increases as Variation Increases  3.Sample Size Width Decreases as Sample Size Increases Width Decreases as Sample Size Increases  4.Distance of X p from Mean  X Width Increases as Distance Increases Width Increases as Distance Increases

37 EPI 809/Spring Why Distance from Mean? Greater dispersion than X 1 XXXX

38 EPI 809/Spring Confidence Interval Estimate Example  Reconsider the Obstetrics example with the following data: Estriol (mg/24h) B.w. (g/1000)  Estimate the mean BW and a subject’s BW response when the Estriol level is 4 at.05 level.

39 EPI 809/Spring Solution Table

40 EPI 809/Spring Confidence Interval Estimate Solution - Mean BW X to be predicted

41 EPI 809/Spring Prediction Interval of Individual Response Note!

42 EPI 809/Spring Why the Extra ‘S’?

43 EPI 809/Spring SAS codes for computing mean and prediction intervals  Data BW; /*Reading data in SAS*/  input estriol birthw;  cards;  11  21  32  42  54  ;  run;  PROC REG data=BW; /*Fitting a linear regression model*/  model birthw=estriol/CLI CLM alpha=.05;  run;

44 EPI 809/Spring Interval Estimate from SAS- Output The REG Procedure Dependent Variable: y Output Statistics Dep Var Predicted Std Error Obs y Value Mean Predict 95% CL Mean 95% CL Predict Residual Predicted Y when X = 3 Confidence Interval SYSYSYSY^ Prediction Interval

45 EPI 809/Spring Hyperbolic Interval Bands

46 EPI 809/Spring Correlation Models

47 EPI 809/Spring Types of Probabilistic Models

48 EPI 809/Spring  Both variables are treated the same in correlation; in regression there is a predictor and a response  In regression the x variable is assumed non- random or measured without error  Correlation is used in looking for relationships, regression for prediction Correlation vs. regression

49 EPI 809/Spring Correlation Models  1.Answer ‘How Strong Is the Linear Relationship Between 2 Variables?’  2.Coefficient of Correlation Used Population Correlation Coefficient Denoted  (Rho) Population Correlation Coefficient Denoted  (Rho) Values Range from -1 to +1 Values Range from -1 to +1 Measures Degree of Association Measures Degree of Association  3.Used Mainly for Understanding

50 EPI 809/Spring  1.Pearson Product Moment Coefficient of Correlation between x and y: Sample Coefficient of Correlation

51 EPI 809/Spring Coefficient of Correlation Values

52 EPI 809/Spring Coefficient of Correlation Values No Correlation

53 EPI 809/Spring Coefficient of Correlation Values Increasing degree of negative correlation No Correlation

54 EPI 809/Spring Coefficient of Correlation Values Perfect Negative Correlation No Correlation

55 EPI 809/Spring Coefficient of Correlation Values Perfect Negative Correlation No Correlation Increasing degree of positive correlation

56 EPI 809/Spring Coefficient of Correlation Values Perfect Positive Correlation Perfect Negative Correlation No Correlation

57 EPI 809/Spring Coefficient of Correlation Examples r = 1r = -1 r =.89r = 0

58 EPI 809/Spring Test of Coefficient of Correlation  1.Shows If There Is a Linear Relationship Between 2 Numerical Variables  2.Same Conclusion as Testing Population Slope  1  3.Hypotheses H 0 :  = 0 (No Correlation) H 0 :  = 0 (No Correlation) H a :   0 (Correlation) H a :   0 (Correlation)

59 EPI 809/Spring Sample t-Test on Correlation Coefficient  Hypotheses H 0 :  = 0 (No Correlation) H 0 :  = 0 (No Correlation) H a :   0 (Correlation) H a :   0 (Correlation)  test statistic: under H 0 t = r (n-2) 1/2 / (1-r 2 ) 1/2 ~ t (n-2) t = r (n-2) 1/2 / (1-r 2 ) 1/2 ~ t (n-2) Reject H 0 if |t| > t α/2, n-2 Reject H 0 if |t| > t α/2, n-2

60 EPI 809/Spring Sample Z-Test on Correlation Coefficient  Hypotheses (Fisher) H 0 :  =  0 H 0 :  =  0 H a :    0 H a :    0  test statistic: under H 0 : Reject H 0 if |z| > z 1- α/2 Reject H 0 if |z| > z 1- α/2

61 EPI 809/Spring Conclusion 1. Describe the Linear Regression Model 2. State the Regression Modeling Steps 3. Explain Ordinary Least Squares 4. Compute Regression Coefficients 5. Understand and check model assumptions 6. Predict Response Variable 7. Comments of SAS Output

62 EPI 809/Spring Conclusion … 8. Correlation Models 9. Test of coefficient of Correlation


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