From last time

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From last time.

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Basic Biostats Topics

• Summary Statistics
• mean, median, mode
• standard deviation, standard error
• Confidence Intervals
• Hypothesis Tests
• t-test (paired and unpaired)
• Chi-Square test
• Fishers exact test

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More Advanced

• Linear Regression
• Logistic Regression
• Repeated Measures Analysis
• Survival Analysis
• Analyzing fMRI data

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General Biostatistics References

• Practical Statistics for Medical Research.
  Altman. Chapman and Hall, 1991.
• Medical Statistics A Common Sense Approach.
  Campbell and Machin. Wiley, 1993
• Principles of Biostatistics. Pagano and
  Gauvreau. Duxbury Press, 1993.
• Fundamentals of Biostatistics. Rosner. Duxbury
  Press, 1993.

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Lecture 3Linear Regression
Child Psychiatry Research Methods Lecture Series

• Elizabeth Garrett
• esg_at_jhu.edu

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Introduction

• Simple linear regression is most useful for
  looking at associations between continuous
  variables.
• We can evaluate if two variables are associated
  linearly.
• We can evaluate how well we can predict one of
  the variables if we know the other.

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Motivating Example (Tierney et al. 2001)

• Is there an association between total sterol
  level and ADI scores in autistic children?
• Hypothesis Children with lower sterol levels
  will tend to have poorer performance (i.e. higher
  scores) on the following components of the ADI
• social
• nonverbal
• repetitive

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Preliminary Data

• 9 individuals with autism
• Some have been on cholesterol supplementation (7
  out of 9)
• Mean age 14
• Age range 8 - 32 years
• Sterol is a continuous variable
• ADI scores are continuous variables

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Statistical Language

• Need to choose what variable is the predicted (Y)
  and which is the predictor (X).
• Y outcome, dependent variable, endogenous
  variable
• X covariate, predictor, regressor, explanatory
  variable, exogenous variable, independent
  variable.
• Our example?

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How can we conclude if there is or is not
an association between sterol and the ADI scores?
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One approach Correlation

• Correlation is a measure of LINEAR association
  between two variables.
• It takes values from -1 to 1.
• Often notated r or ?
• r 1 ? perfect positive correlation
• r -1 ? perfect negative correlation
• r 0 ? no correlation

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r 0.95
r 0.77
r -0.95
r 0.09
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Correlation between ADI measures and Sterol
r -0.85
r -0.70
r 0.06
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Related to r R2

• R2 of variation in Y explained by X.
• Example
• Correlation between nonverbal score and sterol is
  -0.85.
• R2 is 0.852 0.73
• 73 of the variation in nonverbal score is
  explained by sterol
• Gives a sense of the value of sterol in
  predicting nonverbal score
• Other examples
• R2 between sterol and social is 0.49
• R2 between sterol and repetitive is 0.004

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Simple Linear Regression (SLR) Approach

• (1) Fits best line to describe the association
  between Y and X (note straight line)
• (2) Line can be described by two numbers
• - intercept
• - slope
• (3) By-product of regression correlation
  measures how close points fall from the line.
• (4) Why simple? Only one X variable.

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Intercept 24.8
Slope -0.01
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SLR answers two questions.

• Association?
• Does nonverbal score tend to decrease on average
  when sterol increases?
• Is slope different than zero?
• Prediction?
• Can we predict nonverbal score if we know sterol
  level?
• Is the correlation (or R2) high?
• You CAN have association with low correlation!

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Equation of a line

• ?0 Intercept
• ?0 is the estimated nonverbal score if it were
  possible to have a sterol level of 0 (nonsensical
  in this case).
• ?0 calibrates height of line
• ?1 Slope
• ?1 is the estimated change in nonverbal score for
  a one unit change in sterol
• ?1 the estimated difference in nonverbal score
  comparing two kids whose sterol levels differ by
  one.
• We usually use ?1 as our measure of association

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The slope, ?1

• Is ?1 different than zero?

Are each of these reasonable given the data that
we have observed?
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Evaluating Association

• ?1is a statistic, similar to a sample mean, and
  as such has a precision estimate.
• The precision estimate is called the standard
  error of ?1. Denoted se(?1).
• We look at how large ?1 is compared to its
  standard error
• ?1 is often called a regression coefficient or
  a slope.

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General Rule

• If , then we say that
  ?1 is
• statistically significantly different than
  zero.
• T-test interpretation
• H0 ?1 0
• Ha ?1 ? 0
• If is true, then p-value less than
  0.05.
• Intuition
• ?1 is large compared to its precision ? not
  likely that ?1 is 0.

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For large samples.
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ADI Nonverbal and Sterol
Outcome
pvalue

• --------------------------------------------------
  ----------------------------
• nonvrb Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• totster -.0099066 .0022804 -4.344
  0.003 -.0152988 -.0045144
• _cons 24.84349 2.578369 9.635
  0.000 18.74661 30.94036
• --------------------------------------------------
  ----------------------------

se(?1)
?1
Predictor
?0
R-squared 0.73
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Interpretation

• Comparing two autistic kids whose sterol levels
  differ by 1, we estimate that the one with lower
  sterol will have an ADI nonverbal score that is
  higher by 0.01 points.
• Put it in real units
• Comparing two autistic kids whose sterol levels
  differ by 200, we estimate that the child with
  the lower sterol level will have an ADI nonverbal
  score that is higher by 2 points.

(Note 200 x 0.01 2.0)
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A few other details...

• 95 Confidence interval interpretation
• ?1 ? 2se(?1) does not include zero.
• ?1/se(?1) is called the
• t-statistic
• Z-statistic
• If you have small sample (i.e. fewer than 50
  individuals), need to use a t-correction.

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Relationship between correlation and SLR

• Testing that correlation is equal to zero is
  equivalent to testing that the slope is equal to
  zero.
• Can have strong association and low correlation

r 0.93 ?1 1.86 pvalue lt 0.001
r 0.55 ?1 1.88 pvalue lt 0.001
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Additional Points

• (1) Association measured is LINEAR

r 0.02
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Additional Points

• (2) Difference (i.e. distance) between observed
  data and fitted line is called a residual, ?.

1. 0.74 2. -0.95 3. -2.53 4. 3.01
5. 2.52 6. 0.45 7. -3.15 8.
-0.07 9. 0.59 .
?3
?5
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Additional Points

• (3) Often see model equation as

Refers to regression line
Refer to observed data
Generically,
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Additional Points

• (4) Spread of points around line is assumed to be
  constant (i.e. variance of residuals is constant)

BAD!
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Multiple Linear Regression

• More than one X variable
• Generally the same, except
• Cant make plots in multi-dimensions
• Interpretation of ?s is somewhat different

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Other ADI and Sterol SLRs

• How is age when supplementation began related to
  sterol?
• How is age when supplementation began related to
  nonverbal score?

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How might this change our previous result?

• What if age when cholesterol supplementation
  began is associated with both sterol level and
  nonverbal score?
• Is it correct to conclude that total sterol level
  is associated with nonverbal score?

Sterol
Nonverbal Score
Supplementation Age
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We can adjust!

• ------------------------------------------------
  ------------------------------
• nonvrb Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• sterol -.0105816 .0022118 -4.784
  0.003 -.0159937 -.0051696
• agester .1570626 .1158509 1.356
  0.224 -.1264143 .4405394
• _cons 23.81569 2.551853 9.333
  0.000 17.57153 30.05985
• --------------------------------------------------
  ----------------------------

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Interpretation of Betas

• Now that we have adjusted for age at
  supplementation, we need to include that in our
  result
• Comparing two kids who began cholesterol
  supplementation at the same age and whose sterol
  levels differ by 250 units, we estimate that the
  child with the lower sterol level will have an
  ADI nonverbal score higher by 2 points.
• Adjusting for age at supplementation, comparing
  two kids whose sterol levels differ by 250 units,
  we estimate
• Controlling for age at supplementation ..
• Holding age at supplementation constant..

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Collinearity

• If two variables are
• correlated with each other
• correlated with the outcome
• Then, when combined in a MLR model, it could
  happen that
• neither is significant
• only one is significant
• both remain significant

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ADI and Sterol

• Correlation Matrix
• nonvrb sterol agester
• ------------------------------------
• nonvrb 1.0000
• sterol -0.8541 1.0000
• agester 0.0531 0.2251 1.0000

We say that cholesterol time and sterol are
collinear.
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Summing up example.

• After adjusting for age at supplementation, it
  appears that sterol is still a significant
  predictor of ADI nonverbal score.
• BUT!
• Only NINE observations! With more, we would
  almost CERTAINLY see even stronger associations!
• We havent controlled for other potential
  confounders
• length of time on supplementation
• nonverbal score prior to supplementation