EP520 Introductory Biostatistics

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EP520Introductory Biostatistics

• Lecture 25
• Regression and
  Correlation Methods (Chp. 11)
• Multiple linear regression
• Estimation of parameters for multiple linear
  regression
• Inferences for multiple linear regression
• Multiple R-squared (Multiple correlation
  coefficient, for
• assessing goodness of fit)

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Multiple Regression
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Partial Regression Coefficients
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Systolic BP example for multiple linear regression
. list --------------------------
brthwgt agedys sysbp
-------------------------- 1. 135
3 89 2. 120 4 90
3. 100 3 83 4. 105
2 77 5. 130 4 92
-------------------------- 6.
125 5 98 7. 125 2
82 8. 105 3 85 9.
120 5 96 10. 90 4
95 -------------------------- 11.
120 2 80 12. 95
3 79 13. 120 3 86
14. 150 4 97 15. 160
3 92 --------------------------
16. 125 3 88
--------------------------
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. regress sysbp brthwgt agedys Source
SS df MS Number of
obs 16 -----------------------------------
-------- F( 2, 13) 48.08
Model 591.03564 2 295.51782
Prob gt F 0.0000 Residual
79.9018602 13 6.14629694 R-squared
0.8809 ------------------------------------
------- Adj R-squared 0.8626
Total 670.9375 15 44.7291667
Root MSE 2.4792 -------------------------
--------------------------------------------------
--- sysbp Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- brthwgt .1255833 .0343362
3.66 0.003 .0514044 .1997621
agedys 5.887719 .6802051 8.66 0.000
4.418225 7.357213 _cons 53.45019
4.531889 11.79 0.000 43.65964
63.24074 -----------------------------------------
-------------------------------------
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. graph twoway scatter sysbp agedys
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. graph twoway scatter sysbp brthwgt
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. regress sysbp brthwgt agedys Source
SS df MS Number of
obs 16 -----------------------------------
-------- F( 2, 13) 48.08
Model 591.03564 2 295.51782
Prob gt F 0.0000 Residual
79.9018602 13 6.14629694 R-squared
0.8809 ------------------------------------
------- Adj R-squared 0.8626
Total 670.9375 15 44.7291667
Root MSE 2.4792 -------------------------
--------------------------------------------------
--- sysbp Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- brthwgt .1255833 .0343362
3.66 0.003 .0514044 .1997621
agedys 5.887719 .6802051 8.66 0.000
4.418225 7.357213 _cons 53.45019
4.531889 11.79 0.000 43.65964
63.24074 -----------------------------------------
-------------------------------------
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. regress sysbp brthwgt agedys Source
SS df MS Number of
obs 16 -----------------------------------
-------- F( 2, 13) 48.08
Model 591.03564 2 295.51782
Prob gt F 0.0000 Residual
79.9018602 13 6.14629694 R-squared
0.8809 ------------------------------------
------- Adj R-squared 0.8626
Total 670.9375 15 44.7291667
Root MSE 2.4792 -------------------------
--------------------------------------------------
--- sysbp Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- brthwgt .1255833 .0343362
3.66 0.003 .0514044 .1997621
agedys 5.887719 .6802051 8.66 0.000
4.418225 7.357213 _cons 53.45019
4.531889 11.79 0.000 43.65964
63.24074 -----------------------------------------
-------------------------------------
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Systolic BP example including interaction term
. generate age_bwt agedys brthwgt . regress
sysbp brthwgt agedys age_bwt Source
SS df MS Number of obs
16 ---------------------------------------
---- F( 3, 12) 46.87
Model 618.183307 3 206.061102
Prob gt F 0.0000 Residual
52.7541926 12 4.39618272 R-squared
0.9214 ------------------------------------
------- Adj R-squared 0.9017
Total 670.9375 15 44.7291667
Root MSE 2.0967 -------------------------
--------------------------------------------------
--- sysbp Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- brthwgt .551233 .173731
3.17 0.008 .1727057 .9297603
agedys 21.2873 6.223628 3.42 0.005
7.727175 34.84742 age_bwt -.1282085
.0515927 -2.49 0.029 -.2406194
-.0157976 _cons 2.551581 20.83776
0.12 0.905 -42.85 47.95316 -----------
--------------------------------------------------
-----------------
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Regression of FEV on Height (in.) for Boys in
Tecumseh, Michigan Unrestricted Ages 3 - 19
. regress fev hgt if sex1 Source
SS df MS Number of obs
336 -----------------------------------------
-- F( 1, 334) 1174.58 Model
262.711233 1 262.711233 Prob gt
F 0.0000 Residual 74.7035646 334
.223663367 R-squared
0.7786 ------------------------------------------
- Adj R-squared 0.7779 Total
337.414798 335 1.00720835 Root
MSE .47293 ------------------------------
------------------------------------------------
fev Coef. Std. Err. t
Pgtt 95 Conf. Interval ------------------
--------------------------------------------------
--------- hgt .1398832 .0040815
34.27 0.000 .1318544 .1479119
_cons -5.863848 .2544698 -23.04 0.000
-6.364414 -5.363283 ----------------------------
--------------------------------------------------
. graph twoway scatter fev hgt if sex1
lfit fev hgt
Least Squares Regression Equation FEV -5.8638
0.1399 height(in)
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. graph twoway scatter fev height if sex1
lfit fev height Least Squares Regression
Equation FEV -5.8638 0.1399 height(in)
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. twoway lfitci fev hgt
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. summarize age if sex1 Variable
Obs Mean Std. Dev. Min
Max ---------------------------------------------
------------------------ age 336
10.01488 2.975986 3 19 .
generate agelinage-10.01488 if sex1 (318
missing values generated) . generate
agequadagelin2 if sex1 (318 missing values
generated) . summarize if sex1 Variable
Obs Mean Std. Dev. Min
Max -------------------------------------------
-------------------------- id
336 36232.54 23700.52 201
90001 age 336 10.01488
2.975986 3 19 fev
336 2.812446 1.003598 .796
5.793 hgt 336 62.0253
6.330696 47 74 sex
336 1 0 1
1 -----------------------------------------------
---------------------- smoke 336
.077381 .2675934 0 1
agelin 336 9.47e-07 2.975986
-7.01488 8.98512 agequad 336
8.830136 12.62822 .0002214 80.73238
Center age at 0
Compute quadratic term for age
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. generate ht_linhgt-62.0253 if sex1 (318
missing values generated) . generate
ht_quadht_lin2 if sex1 (318 missing values
generated) . summarize if sex1 Variable
Obs Mean Std. Dev. Min
Max -------------------------------------------
-------------------------- id
336 36232.54 23700.52 201
90001 age 336 10.01488
2.975986 3 19 fev
336 2.812446 1.003598 .796
5.793 hgt 336 62.0253
6.330696 47 74 sex
336 1 0 1
1 -----------------------------------------------
---------------------- smoke 336
.077381 .2675934 0 1
agelin 336 9.47e-07 2.975986
-7.01488 8.98512 agequad 336
8.830136 12.62822 .0002214 80.73238
ht_lin 336 -2.41e-06 6.330696
-15.0253 11.9747 ht_quad 336
39.95844 41.29339 .0006401 225.7596
Center height at 0
Compute quadratic term for height
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. regress fev ht_lin if sex1 Source
SS df MS Number of
obs 336 -----------------------------------
-------- F( 1, 334) 1174.58
Model 262.711233 1 262.711233
Prob gt F 0.0000 Residual
74.7035646 334 .223663367 R-squared
0.7786 ------------------------------------
------- Adj R-squared 0.7779
Total 337.414798 335 1.00720835
Root MSE .47293 -------------------------
--------------------------------------------------
--- fev Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- ht_lin .1398832 .0040815
34.27 0.000 .1318544 .1479119
_cons 2.812447 .0258005 109.01 0.000
2.761695 2.863199 ----------------------------
--------------------------------------------------
. regress fev ht_lin ht_quad if sex1
Source SS df MS
Number of obs 336 -------------------------
------------------ F( 2, 333)
680.88 Model 271.116793 2
135.558397 Prob gt F 0.0000
Residual 66.2980047 333 .199093107
R-squared 0.8035 ------------------------
------------------- Adj R-squared
0.8023 Total 337.414798 335
1.00720835 Root MSE
.4462 -------------------------------------------
----------------------------------- fev
Coef. Std. Err. t Pgtt 95
Conf. Interval ---------------------------------
--------------------------------------------
ht_lin .1450898 .0039333 36.89 0.000
.1373525 .152827 ht_quad .0039182
.000603 6.50 0.000 .002732
.0051044 _cons 2.655882 .0342511
77.54 0.000 2.588506 2.723258 -----------
--------------------------------------------------
-----------------
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. regress fev ht_lin ht_quad agelin if sex1
Source SS df MS
Number of obs 336 -----------------------
-------------------- F( 3, 332)
501.58 Model 276.425205 3
92.1417351 Prob gt F 0.0000
Residual 60.9895927 332 .183703592
R-squared 0.8192 ------------------------
------------------- Adj R-squared
0.8176 Total 337.414798 335
1.00720835 Root MSE
.42861 ------------------------------------------
------------------------------------ fev
Coef. Std. Err. t Pgtt 95
Conf. Interval ---------------------------------
--------------------------------------------
ht_lin .1146489 .0068076 16.84 0.000
.1012575 .1280402 ht_quad .0037141
.0005805 6.40 0.000 .0025722
.004856 agelin .0769142 .0143081
5.38 0.000 .0487681 .1050602
_cons 2.664039 .0329357 80.89 0.000
2.59925 2.728828 ----------------------------
--------------------------------------------------

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. regress fev ht_lin ht_quad agelin agequad if
sex1 Source SS df MS
Number of obs
336 -------------------------------------------
F( 4, 331) 383.92 Model
277.584619 4 69.3961548 Prob gt F
0.0000 Residual 59.8301789 331
.180755827 R-squared
0.8227 ------------------------------------------
- Adj R-squared 0.8205 Total
337.414798 335 1.00720835 Root
MSE .42515 ------------------------------
------------------------------------------------
fev Coef. Std. Err. t
Pgtt 95 Conf. Interval ------------------
--------------------------------------------------
--------- ht_lin .1218146 .0073215
16.64 0.000 .1074121 .1362171
ht_quad .0027855 .0006826 4.08 0.000
.0014427 .0041283 agelin .0543468
.0167582 3.24 0.001 .0213808
.0873127 agequad .0063499 .0025072
2.53 0.012 .0014178 .011282
_cons 2.645071 .0335178 78.92 0.000
2.579136 2.711006 ----------------------------
--------------------------------------------------

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Goodness of fit criteria Plot of studentized
residuals versus predicted values of systolic
blood pressure . predict yhat (option xb
assumed fitted values) . label variable yhat
"Predicted value of sbp" . predict e, rstudent .
label variable e "Studentized residual" . graph
twoway scatter e yhat
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Plot of studentized residual versus age . graph
twoway scatter e agedys
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. graph twoway scatter e brthwgt
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Standardized Regression Coefficient
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Standardized Regression Coefficient
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This shows equivalently of t-test and regression
with one indicator variable (2 groups). Recall
that two-sample t-test with independent samples
is a special case of one way (fixed effects)
ANOVA.
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Dummy variables
Extending this relationship Multiple-group
analysis Goal to show that one-way ANOVA is a
special case of regression. Use of dummy
variables to represent a categorical variable
with k categories.
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Use of dummy variables to represent a categorical
variable with k categories. Categorical
variable C with k categories
Category k is reference (or baseline) group.
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Treatment Category Dummy variable x1
x2 x3 x4 1 1 0 0 0 2 0 1 0 0 3 0 0 1 0 4
0 0 0 1 5 0 0 0 0
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Anova and Regression are the same test
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EP520 Introductory Biostatistics

• Lecture 26
• Regression and Correlation Methods
  (chapter 11)
• Relationship between Regression and ANOVA
• Analysis of covariance
• Two-way ANOVA (brief)
• Testing for correlations

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Relationship between Multiple Linear Regression
and One Way ANOVA

• E(y) µ aj
• Want to compare underlying means among k
• groups where observations in group j are
• N(µj µ aj, s2).
• To test H0 aj 0, for all j 1,,k vs.
• H0 at least one of the ais are different.

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• Two Equivalent Methods
• Overall F Test for One-Way ANOVA (Fixed)
• Set up Multiple Regression Model
• Between SS Regression SS
• Within SS Residual SS
• F Statistics and p-values are same

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• . Comparing ANOVA and Regression - Aspirin
  Example
• . list
• --------------------------------------
• drug fevreduc ind1 ind2 ind5
• --------------------------------------
• 1. 1 2 1 0 0
• 2. 1 1.6 1 0 0
• 3. 1 2.1 1 0 0
• 4. 1 .6 1 0 0
• 5. 1 1.3 1 0 0
• --------------------------------------
• 6. 2 .5 0 1 0
• 7. 2 1.2 0 1 0
• 8. 2 .3 0 1 0
• 9. 2 .2 0 1 0
• 10. 2 -.4 0 1 0
• --------------------------------------
• 11. 3 1.1 0 0 1

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• . anova fevreduc drug, detail reg
• Factor Value Value
  Value Value
• ------------------------------------------------
  ------------------------
• drug 1 1 2 2 3 3
  
• Source SS df MS
  Number of obs 15
• -------------------------------------------
  F( 2, 12) 6.79
• Model 5.82933309 2 2.91466655
  Prob gt F 0.0106
• Residual 5.14800001 12 .429000001
  R-squared 0.5310
• -------------------------------------------
  Adj R-squared 0.4529
• Total 10.9773331 14 .784095222
  Root MSE .65498
• --------------------------------------------------
  ----------------------------
• fevreduc Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• --------------------------------------------------
  ----------------------------
• _cons .08 .2929164 0.27 0.789
  -.5582099 .71821
• Drug 1 1.44 .4142463 3.48 0.005
  .5374348 2.342565

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• . regress fevreduc ind1 ind2
• Source SS df MS
  Number of obs 15
• -------------------------------------------
  F( 2, 12) 6.79
• Model 5.82933309 2 2.91466655
  Prob gt F 0.0106
• Residual 5.14800001 12 .429000001
  R-squared 0.5310
• -------------------------------------------
  Adj R-squared 0.4529
• Total 10.9773331 14 .784095222
  Root MSE .65498
• --------------------------------------------------
  ----------------------------
• fevreduc Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• ind1 1.44 .4142463 3.48
  0.005 .5374348 2.342565
• ind2 .28 .4142463 0.68
  0.512 -.6225652 1.182565
• _cons .08 .2929164 0.27
  0.789 -.5582099 .71821
• --------------------------------------------------
  ----------------------------

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• . regress fevreduc ind1 ind2 ind3
• Source SS df MS
  Number of obs 15
• -------------------------------------------
  F( 2, 12) 6.79
• Model 5.82933309 2 2.91466655
  Prob gt F 0.0106
• Residual 5.14800001 12 .429000001
  R-squared 0.5310
• -------------------------------------------
  Adj R-squared 0.4529
• Total 10.9773331 14 .784095222
  Root MSE .65498
• --------------------------------------------------
  ----------------------------
• fevreduc Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• ind1 1.16 .4142463 2.80
  0.016 .2574348 2.062565
• ind2 (dropped)
• ind3 -.28 .4142463 -0.68
  0.512 -1.182565 .6225652
• _cons .36 .2929164 1.23
  0.243 -.2782099 .99821
• --------------------------------------------------
  ----------------------------

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Analysis of Covariance

• Say we want to compare k groups using
• ANOVA (regression with dummies) but also
• need to adjust for differences in age in the
• groups.
• y a ß1x1 ß2x2 ßk-1xk-1 age ßAGE
• This is called an Analysis of Covariance Model

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Two Way ANOVAStudy performed to look at level of
Systolic BP for 3 Groups
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Correlation

• Problem Interested in the relationship
• between two variables, Height and Weight,
• Age and Systolic BP, REM Sleep and
• Stress, etc. Want to quantify this
• relationship. The Correlation Coefficient
• (Pearson Correlation) is useful for this
• purpose.

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• Note Plot of x vs. y
• looking like a straight
• Line is indicative of high
• correlation, does not
• need to be 45 degree
• Line.

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• One Sample T test for Correlation Coefficient
• . display 2(1-normprob(0.25(100-2)0.5/(1-(0.25)
  2)0.5))
• .01058714
• One Sample Z test for Correlation Coefficient
• . display 0.5ln((10.38)/(1-0.38))
• .40005965
• . display 0.5ln((10.5)/(1-0.5))
• .54930614
• . display 2normprob(-1.4699075)
• .14158681

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• Two Sample tests for Correlations
• Example
• Two groups of children. One group lives with
• natural parents, other group lives with adopted
• parents.
• Is correlation between BP (SBP?) of mother
• and child different for these groups? Would
• suggest genetic link.

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