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Acknowledgments• Funding
– National Institutes of Health– United States Olympic Foundation
• Collaborators– Walter “Buzz” Stewart– Charlie Hall, Scott Zeger, David Simon
• References– Stewart WF et al., A prospective study of CNS function in US amateur
boxers, Am J Epidemiol 1994; 139: 573-88.– Bandeen-Roche K et al., Modelling disease progression in terms of
exposure history, Statist Med 1999; 18:2899-2916.
Introduction
(Imagine: 1989 news photo of Larry Holmes pounding the face of James “Bonecrusher” Smith)
• Well publicized: Boxing may cause neurological harm
• ~ 1986: IOC explores eliminating boxing (for golf?)
• Olympic boxing is amateur: different from pro
• Research study initiated: NIH / USABF collaboration
Scientific Question: Does boxing cause cerebral injury?
• Hypothesized pathway: brain jarring NEURO- PSYCHOLOGIC
OUTCOMES BOXING BRAIN CEREBRAL ELECTRO-BOUTS* JARRING INJURY PHYSIOLOGICSPAR OUTCOMES
NEUROLOGICOUTCOMES
---------------------------------------------------
SYMPTOMSEXPOSURE DOSE PHYSIOLOGIC
IMPAIRMENT SIGNS
Scientific Question: Does boxing cause cerebral injury?
• Injury model
– Mild, transient• Focal axonal damage, re-growth• No measurable long-term injury
– Cell disruption sufficient to cause hemorrhage• Progressive axonal death• Measurable long-term injury
Brief Study Design• "Full" Boxing club sample
– NY, DC, Cleveland, St. Louis, Louisiana, Houston
• N = 593 boxers – One baseline and three follow-up exams “per boxer”; 1988-1994– N=493 with a first follow up
• Outcomes – 17 neuropsychological tests (Today: Block
Design)– Electrophysiologic Battery– Ataxia and Neurological Tests
• Covariates– Primary: number of bouts boxed– Secondary: age, race, education, Ravens IQ score, club,
non-boxing concussion history, drug test result
Step 1:Formulate model
• Question: Do blocks scores tend to decrease as # of bouts increases?– Critique an approach: “Pool” all four
rounds of data, and regress bouts (Y) on blocks score (X)
• Wrong direction: Should be blocks (Y) on bouts (X)
• Independence assumption violated: Multiple measures on same person; also clustering within clubs
• Weak causal content: Fails to use within-person change
UnlinkingEffect evidence: Status
versus Change Association: Reaction Time (sec) & Bouts Boxed
0
0.1
0.2
0.3
0.4
0 20 40 60 80 100
Bouts boxed
Rea
ctio
n t
ime
UnlinkingEffect evidence: Status
versus Change Association: Reaction Time (sec) & Bouts Boxed
0
0.1
0.2
0.3
0.4
0 20 40 60 80 100
Bouts boxed
Rea
ctio
n t
ime
Model Building
• Suppose goal = capture both relationships: status and change– Considered, rejected: E[Yit|Xi] = 0+ 1Xit
• Y = blocks score; X=#bouts• i=people 1,…,n; t=times 1, 2 (…)
– Way to think: status 1 & change 2 Time 1: E[Yi1|Xi] = 0+ 1Xi1
+ 3Time 2: E[Yi2|Xi] = 0+ 1Xi1 + 2(Xi2-Xi1)
Allows age-related change between t1 and t2
Model BuildingE[Yi1|Xi] = 0+ 1Xi1
E[Yi2|Xi] = 0+ 1Xi1 +2(Xi2-Xi1) + 3
i.e.
E[Yit|Xi1,Xi2] = 0+ 1Xi1 +2(Xi2-Xi1)*1{t=2} + 31{t=2}
• Interpret 3
• How to test for equal status, change relationships?f
• Zero out other coefficients you can: Xi1 = Xi2-Xi1=0• Then, time 2 mean = 0+ 3; time 1 mean = 0
• 3 = Mean change in block score among non-boxers thru time 2
• Test H0: 2 = 1
Model Building
• From now on: we’ll analyze relationship between change in blocks score (t2-t1) and – baseline bout total– change in bout total– N=413 in the analysis
• Why the baseline bout total?• Models potentially delayed effect
Exploratory Data Analysis
blkdiff
blbouts
boutdiff
-20
0
20
-20 0 20
0
200
400
0 200 400
-500
0
500
1000
-500 0 500 1000
blkdiff
blbouts
boutdiffy=0
New model building goal
• From now on: we’ll analyze relationship between change in blocks score (t2-t1) and – baseline bout total– change in bout total
• In real life: validation, errors-in-variables (covariates) analysis
Exploratory Data AnalysisScatterplot: Blocks Change vs. BL Bouts
-20
-10
010
20
blk
diff
0 100 200 300 400blbouts
bandwidth = .8
Lowess smoother
Exploratory Data AnalysisScatterplot: Blocks Change vs. BL Bouts
-20
-10
010
20
blk
diff
0 20 40 60 80blbouts
bandwidth = .8
Lowess smoother
.lowess blkdiff blbouts if blbouts < 75
Modeling options• Linear Y, X model
OTHERS?
• Polynomial Y, X model
• Replace X by √X, etc. (transform)
• Categorize X
• Spline Y, X model
Highly sensitive to extreme points
Obscure interpretation
Wastes much exposure information; categories arbitrary?
Spline ModelRelationship: Change in Blocks, Bouts
• Choice of knots– Novice versus
Open divisions: 10 bouts
– Median of remaining bouts: 35
– Histogram suggests a cut at around 75:
0.0
1.0
2.0
3.0
4D
ensi
ty
0 100 200 300 400blbouts
Spline ModelRelationship: Change in Blocks, Bouts
• Order– Plot up to 75 bouts appears fairly linear– Smooth after 75 bouts appears fairly linear
• (Population) Model: E[Yi2-Yi1|Xi1] =
0+ 1Xi1 +2(Xi1-10)+ + 3(Xi1-35)+ +4(Xi1-75)+
- Order = 1
- Number of polynomial terms underlying relationship
Aside
• Suppose X =
(0,1,5,11,14,30,36,55,78,102)
• What is the design matrix for the model on the previous slide?
(Posted version of slides will include answer)
Regression model• regress blkdiff blbouts boutspl1 boutspl2 boutspl3
Source | SS df MS Number of obs = 413-------------+------------------------------ F( 4, 408) = 1.92 Model | 281.256924 4 70.314231 Prob > F = 0.1058 Residual | 14922.6559 408 36.575137 R-squared = 0.0185-------------+------------------------------ Adj R-squared = 0.0089 Total | 15203.9128 412 36.9027011 Root MSE = 6.0477
------------------------------------------------------------------------------ blkdiff | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- blbouts | .2049663 .1111387 1.84 0.066 -.0135095 .4234422 boutspl1 | -.3300803 .145362 -2.27 0.024 -.6158321 -.0443284 boutspl2 | .1565677 .0787441 1.99 0.047 .0017729 .3113624 boutspl3 | -.033317 .0469676 -0.71 0.479 -.1256457 .0590117 _cons | 1.452344 .7033785 2.06 0.040 .0696462 2.835043------------------------------------------------------------------------------
Mean Block Score Change, 0 Bouts
Mean per-bout diff in Blocks Change, Novice Boxers
Mean per-10 bout diff in Blocks Change, Novice Boxers? 2.05 points
In each case, coefficient estimates the population mean!
Regression model• regress blkdiff blbouts boutspl1 boutspl2 boutspl3
Source | SS df MS Number of obs = 413-------------+------------------------------ F( 4, 408) = 1.92 Model | 281.256924 4 70.314231 Prob > F = 0.1058 Residual | 14922.6559 408 36.575137 R-squared = 0.0185-------------+------------------------------ Adj R-squared = 0.0089 Total | 15203.9128 412 36.9027011 Root MSE = 6.0477
------------------------------------------------------------------------------ blkdiff | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- blbouts | .2049663 .1111387 1.84 0.066 -.0135095 .4234422 boutspl1 | -.3300803 .145362 -2.27 0.024 -.6158321 -.0443284 boutspl2 | .1565677 .0787441 1.99 0.047 .0017729 .3113624 boutspl3 | -.033317 .0469676 -0.71 0.479 -.1256457 .0590117 _cons | 1.452344 .7033785 2.06 0.040 .0696462 2.835043------------------------------------------------------------------------------
Boxed t-test, CI tests H0: 4=0, i.e. no difference in per-bout difference in mean serial test performance change, above 75 bouts versus on range of 35-75 bouts
Regression model
Notice that effect attenuates a little bit, but standard error decreases, and t statistic increases.
• regress blkdiff blbouts boutspl1 boutspl2
Source | SS df MS Number of obs = 413-------------+------------------------------ F( 3, 409) = 2.40 Model | 262.852541 3 87.6175137 Prob > F = 0.0675 Residual | 14941.0603 409 36.5307098 R-squared = 0.0173-------------+------------------------------ Adj R-squared = 0.0101 Total | 15203.9128 412 36.9027011 Root MSE = 6.0441
------------------------------------------------------------------------------ blkdiff | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- blbouts | .1943893 .110067 1.77 0.078 -.0219783 .4107569 boutspl1 | -.300474 .1391568 -2.16 0.031 -.5740259 -.0269222 boutspl2 | .1117487 .0469677 2.38 0.018 .0194205 .2040768 _cons | 1.480605 .7018227 2.11 0.035 .1009757 2.860235------------------------------------------------------------------------------
What is good, bad about the estimates?
• The good– Accuracy (estimator is unbiased if correct
mean model; SEs are accurate if correct A1-A4)
– Precision (estimator is BLUE)
• The bad– Not terribly robust (may be influenced by
isolated points)
The Estimated Relationship:Mean Block Score Change, Bouts
-20
-10
010
20
blk
diff
/Fitt
ed v
alu
es
0 100 200 300 400blbouts
blkdiff Fitted values
Slope = .19Slope = .19-.30 = -.11
Slope ≈ .19-.30+.11≈0
Estimated Relationship:Mean Block Score Change, Bouts
-20
-10
010
20
blk
diff/F
itte
d v
alu
es
0 20 40 60 80blbouts
blkdiff Fitted values On bout range < 75
Comments
• Odd finding: Apparent benefit of novice boxing, and loss of benefit (back to nominal) in early open boxing
• Checked for influence: Little• Are we being misled by relationships
with other variables?– Age– BL blocks design score
Relationship between block score change and baseline block score
-20
-10
010
20
blk
diff
0 10 20 30 40 50blblocks
Regression ModelAdjusting for Baseline Block Score, Age
• regress blkdiff blbouts boutspl1 boutspl2 boutspl3 cenblock cenage
Source | SS df MS Number of obs = 413-------------+------------------------------ F( 6, 406) = 6.99 Model | 1423.72075 6 237.286792 Prob > F = 0.0000 Residual | 13780.1921 406 33.9413598 R-squared = 0.0936-------------+------------------------------ Adj R-squared = 0.0802 Total | 15203.9128 412 36.9027011 Root MSE = 5.8259
------------------------------------------------------------------------------ blkdiff | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- blbouts | .1808688 .1071491 1.69 0.092 -.0297675 .3915052 boutspl1 | -.2856816 .1402603 -2.04 0.042 -.5614087 -.0099546 boutspl2 | .1390732 .0759187 1.83 0.068 -.0101697 .2883161 boutspl3 | -.0364184 .0452629 -0.80 0.422 -.1253974 .0525606 cenblock | -.1591111 .0324602 -4.90 0.000 -.2229221 -.0953 cenage | -.2683838 .1223957 -2.19 0.029 -.5089922 -.0277754
Little change in direct effects (here) from total (slide 22)
. gen cenage=blage-17;
. gen cenblock=blblocks-25If final (blue) spline term removed, RSS = 13802.1648 ; SSreg = 1401.74799
General F-testingIs there evidence of nonlinearity in the Blocks change / Bouts relationship?
• Step 1: Fit model with age, baseline blocks score, baseline bouts only. (Call these variables X1) Save the RSS.
. regress blkdiff blbouts cenblock cenage
Source | SS df MS Number of obs = 413 Model | 1254.61956 3 418.20652 Prob > F = 0.0000
Residual | 13949.2933 409 34.1058515 R-squared = 0.0825-------------+------------------------------ Adj R-squared = 0.0758 Total | 15203.9128 412 36.9027011 Root MSE = 5.84
------------------------------------------------------------------------------ blkdiff | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- blbouts | -.0037381 .0065357 -0.57 0.568 -.0165858 .0091096 cenblock | -.1628229 .0324808 -5.01 0.000 -.2266731 -.0989727 cenage | -.2787612 .1224796 -2.28 0.023 -.5195294 -.0379931 _cons | 1.89059 .3536493 5.35 0.000 1.195393 2.585787------------------------------------------------------------------------------
General F-testingIs there evidence of nonlinearity in the Blocks change / Bouts relationship?
• Step 2: Fit model with age, baseline blocks score, baseline bouts, and spline terms for 10, 35 bouts
– Done on slide 29. Save RSSL = 13802
General F-testingIs there evidence of nonlinearity in the Blocks change / Bouts relationship?
• Sequential ANOVA table:
Source SS df MS
Regression
X1
Splines|X1
Residual
Total
13802
1401.7
15204 (add)
147= 13949-13802
1255
and 147 must add to 1402
5 3 2
411412
SS/df(all cases)
General F-testingIs there evidence of nonlinearity in the Blocks change / Bouts relationship?
• Step 3: F-test
• [(RSSS-RSSL)/(pj)]/[RSSL/(n-p-1)]
– pj = # extra parameters in larger vs. smaller model
– p = number of covariates in larger model
– RSSL/(n-p-1) = residual variance estimate (larger model)
(2)
(5)
General F-testingIs there evidence of nonlinearity in the Blocks change / Bouts relationship?
• Step 3: F-test
• [(RSSS-RSSL)/(pj)]/[RSSL/(n-p-1)]
= [(13949-13802)/2]/[13802/411]
= [147/2]/[13802/411]
= 73.5/33.6
= 2.19
Compare to F2,411(.95) = 3.02; is less – do not reject!