statistics report: regression and hypothesis testing for the dependent and independent things that effects the topic
alsh3
regression.pptLinear correlation and linear regression
Recall: Covariance
cov(X,Y) > 0 X and Y are positively correlated
cov(X,Y) < 0 X and Y are inversely correlated
cov(X,Y) = 0 X and Y are independent
Interpreting Covariance
Correlation coefficient
- Pearson’s Correlation Coefficient is standardized covariance (unitless):
Correlation
- Measures the relative strength of the linear relationship between two variables
- Unit-less
- Ranges between –1 and 1
- The closer to –1, the stronger the negative linear relationship
- The closer to 1, the stronger the positive linear relationship
- The closer to 0, the weaker any positive linear relationship
Scatter Plots of Data with Various Correlation Coefficients
Y
X
Y
X
Y
X
Y
X
Y
X
r = -1
r = -.6
r = 0
r = +.3
r = +1
Y
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r = 0
- Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
Y
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Linear relationships
Curvilinear relationships
Linear Correlation
- Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
Y
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Y
Y
X
X
Strong relationships
Weak relationships
Linear Correlation
- Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
Linear Correlation
Y
X
Y
X
No relationship
- Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
Calculating by hand…
Simpler calculation formula…
Numerator of covariance
Numerators of variance
Distribution of the correlation coefficient:
*note, like a proportion, the variance of the correlation coefficient depends on the correlation coefficient itselfsubstitute in estimated r
The sample correlation coefficient follows a T-distribution with n-2 degrees of freedom (since you have to estimate the standard error).
Linear regression
In correlation, the two variables are treated as equals. In regression, one variable is considered independent (=predictor) variable (X) and the other the dependent (=outcome) variable Y.
What is “Linear”?
- Remember this:
- Y=mX+B?
B
m
What’s Slope?
A slope of 2 means that every 1-unit change in X yields a 2-unit change in Y.
Prediction
If you know something about X, this knowledge helps you predict something about Y. (Sound familiar?…sound like conditional probabilities?)
Regression equation…
Expected value of y at a given level of x=
Predicted value for an individual…
yi= + *xi + random errori
Follows a normal distribution
Fixed – exactly on the line
Assumptions (or the fine print)
- Linear regression assumes that…
- 1. The relationship between X and Y is linear
- 2. Y is distributed normally at each value of X
- 3. The variance of Y at every value of X is the same (homogeneity of variances)
- 4. The observations are independent
The standard error of Y given X is the average variability around the regression line at any given value of X. It is assumed to be equal at all values of X.
Sy/x
Sy/x
Sy/x
Sy/x
Sy/x
Sy/x
*Least squares estimation gave us the line (β) that minimized C2
Regression Picture
R2=SSreg/SStotal
C
A
B
A
yi
x
y
yi
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B
SStotal
Total squared distance of observations from naïve mean of y
Total variation
SSreg
Distance from regression line to naïve mean of y
Variability due to x (regression)
SSresidual
Variance around the regression line
Additional variability not explained by x—what least squares method aims to minimize
A2 B2 C2
Recall example: cognitive function and vitamin D
- Hypothetical data loosely based on [1]; cross-sectional study of 100 middle-aged and older European men.
- Cognitive function is measured by the Digit Symbol Substitution Test (DSST).
1. Lee DM, Tajar A, Ulubaev A, et al. Association between 25-hydroxyvitamin D levels and cognitive performance in middle-aged and older European men. J Neurol Neurosurg Psychiatry. 2009 Jul;80(7):722-9.
Distribution of vitamin D
Mean= 63 nmol/L
Standard deviation = 33 nmol/L
*
Distribution of DSST
Normally distributed
Mean = 28 points
Standard deviation = 10 points
*
Four hypothetical datasets
- I generated four hypothetical datasets, with increasing TRUE slopes (between vit D and DSST):
- 0
- 0.5 points per 10 nmol/L
- 1.0 points per 10 nmol/L
- 1.5 points per 10 nmol/L
Dataset 1: no relationship
Dataset 2: weak relationship
Dataset 3: weak to moderate relationship
Dataset 4: moderate relationship
The “Best fit” line
Regression equation:
E(Yi) = 28 + 0*vit Di (in 10 nmol/L)
The “Best fit” line
Note how the line is a little deceptive; it draws your eye, making the relationship appear stronger than it really is!
Regression equation:
E(Yi) = 26 + 0.5*vit Di (in 10 nmol/L)
The “Best fit” line
Regression equation:
E(Yi) = 22 + 1.0*vit Di (in 10 nmol/L)
The “Best fit” line
Regression equation:
E(Yi) = 20 + 1.5*vit Di (in 10 nmol/L)
Note: all the lines go through the point (63, 28)!
Estimating the intercept and slope: least squares estimation
** Least Squares Estimation
A little calculus….
What are we trying to estimate? β, the slope, from
What’s the constraint? We are trying to minimize the squared distance (hence the “least squares”) between the observations themselves and the predicted values , or (also called the “residuals”, or left-over unexplained variability)
Differencei = yi – (βx + α) Differencei2 = (yi – (βx + α)) 2
Find the β that gives the minimum sum of the squared differences. How do you maximize a function? Take the derivative; set it equal to zero; and solve. Typical max/min problem from calculus….
From here takes a little math trickery to solve for β…
Resulting formulas…
Slope (beta coefficient) =
Intercept=
Regression line always goes through the point:
Relationship with correlation
In correlation, the two variables are treated as equals. In regression, one variable is considered independent (=predictor) variable (X) and the other the dependent (=outcome) variable Y.
Example: dataset 4
SDx = 33 nmol/L
SDy= 10 points
Cov(X,Y) = 163 points*nmol/L
Beta = 163/332 = 0.15 points per nmol/L
= 1.5 points per 10 nmol/L
r = 163/(10*33) = 0.49
Or
r = 0.15 * (33/10) = 0.49
Significance testing…
Slope
Distribution of slope ~ Tn-2(β,s.e.( ))
H0: β1 = 0 (no linear relationship)
H1: β1 0 (linear relationship does exist)
Tn-2=
Formula for the standard error of beta (you will not have to calculate by hand!):
Example: dataset 4
- Standard error (beta) = 0.03
- T98 = 0.15/0.03 = 5, p<.0001
- 95% Confidence interval = 0.09 to 0.21
Residual Analysis: check assumptions
- The residual for observation i, ei, is the difference between its observed and predicted value
- Check the assumptions of regression by examining the residuals
- Examine for linearity assumption
- Examine for constant variance for all levels of X (homoscedasticity)
- Evaluate normal distribution assumption
- Evaluate independence assumption
- Graphical Analysis of Residuals
- Can plot residuals vs. X
Predicted values…
For Vitamin D = 95 nmol/L (or 9.5 in 10 nmol/L):
Residual =
observed - predicted
X=95 nmol/L
34
Residual Analysis for Linearity
Not Linear
Linear
x
residuals
x
Y
x
Y
x
residuals
- Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
Residual Analysis for
Homoscedasticity
Non-constant variance
Constant variance
x
x
Y
x
x
Y
residuals
residuals
- Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
Residual Analysis for Independence
Not Independent
Independent
X
X
residuals
residuals
X
residuals
- Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
*
Residual plot, dataset 4
Multiple linear regression…
- What if age is a confounder here?
- Older men have lower vitamin D
- Older men have poorer cognition
- “Adjust” for age by putting age in the model:
- DSST score = intercept + slope1xvitamin D + slope2 xage
2 predictors: age and vit D…
21.unknown
Different 3D view…
22.unknown
Fit a plane rather than a line…
On the plane, the slope for vitamin D is the same at every age; thus, the slope for vitamin D represents the effect of vitamin D when age is held constant.
23.unknown
Equation of the “Best fit” plane…
- DSST score = 53 + 0.0039xvitamin D (in 10 nmol/L) - 0.46 xage (in years)
- P-value for vitamin D >>.05
- P-value for age <.0001
- Thus, relationship with vitamin D was due to confounding by age!
Multiple Linear Regression
- More than one predictor…
E(y)= + 1*X + 2 *W + 3 *Z…
Each regression coefficient is the amount of change in the outcome variable that would be expected per one-unit change of the predictor, if all other variables in the model were held constant.
Functions of multivariate analysis:
- Control for confounders
- Test for interactions between predictors (effect modification)
- Improve predictions
A ttest is linear regression!
- Divide vitamin D into two groups:
- Insufficient vitamin D (<50 nmol/L)
- Sufficient vitamin D (>=50 nmol/L), reference group
- We can evaluate these data with a ttest or a linear regression…
As a linear regression…
Parameter ````````````````Standard
Variable Estimate Error t Value Pr > |t|
Intercept 40.07407 1.47511 27.17 <.0001
insuff -7.53060 2.17493 -3.46 0.0008
Intercept represents the mean value in the sufficient group.
Slope represents the difference in means between the groups. Difference is significant.
ANOVA is linear regression!
- Divide vitamin D into three groups:
- Deficient (<25 nmol/L)
- Insufficient (>=25 and <50 nmol/L)
- Sufficient (>=50 nmol/L), reference group
DSST= (=value for sufficient) + insufficient*(1 if insufficient) + 2 *(1 if deficient)
This is called “dummy coding”—where multiple binary variables are created to represent being in each category (or not) of a categorical variable
The picture…
Sufficient vs. Insufficient
Sufficient vs. Deficient
Results…
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 40.07407 1.47817 27.11 <.0001
deficient 1 -9.87407 3.73950 -2.64 0.0096
insufficient 1 -6.87963 2.33719 -2.94 0.0041
- Interpretation:
- The deficient group has a mean DSST 9.87 points lower than the reference (sufficient) group.
- The insufficient group has a mean DSST 6.87 points lower than the reference (sufficient) group.
Other types of multivariate regression
Multiple linear regression is for normally distributed outcomes
Logistic regression is for binary outcomes
Cox proportional hazards regression is used when time-to-event is the outcome
Common multivariate regression models.
| Outcome (dependent variable) | Example outcome variable | Appropriate multivariate regression model | Example equation | What do the coefficients give you? |
| Continuous | Blood pressure | Linear regression | blood pressure (mmHg) = + salt*salt consumption (tsp/day) + age*age (years) + smoker*ever smoker (yes=1/no=0) | slopes—tells you how much the outcome variable increases for every 1-unit increase in each predictor. |
| Binary | High blood pressure (yes/no) | Logistic regression | ln (odds of high blood pressure) = + salt*salt consumption (tsp/day) + age*age (years) + smoker*ever smoker (yes=1/no=0) | odds ratios—tells you how much the odds of the outcome increase for every 1-unit increase in each predictor. |
| Time-to-event | Time-to- death | Cox regression | ln (rate of death) = + salt*salt consumption (tsp/day) + age*age (years) + smoker*ever smoker (yes=1/no=0) | hazard ratios—tells you how much the rate of the outcome increases for every 1-unit increase in each predictor. |
Multivariate regression pitfalls
Multi-collinearity
Residual confounding
Overfitting
Multicollinearity
Multicollinearity arises when two variables that measure the same thing or similar things (e.g., weight and BMI) are both included in a multiple regression model; they will, in effect, cancel each other out and generally destroy your model.
Model building and diagnostics are tricky business!
Residual confounding
- You cannot completely wipe out confounding simply by adjusting for variables in multiple regression unless variables are measured with zero error (which is usually impossible).
- Example: meat eating and mortality
Men who eat a lot of meat are unhealthier for many reasons!
Sinha R, Cross AJ, Graubard BI, Leitzmann MF, Schatzkin A. Meat intake and mortality: a prospective study of over half a million people. Arch Intern Med 2009;169:562-71
Mortality risks…
Sinha R, Cross AJ, Graubard BI, Leitzmann MF, Schatzkin A. Meat intake and mortality: a prospective study of over half a million people. Arch Intern Med 2009;169:562-71
Overfitting
- In multivariate modeling, you can get highly significant but meaningless results if you put too many predictors in the model.
- The model is fit perfectly to the quirks of your particular sample, but has no predictive ability in a new sample.
Overfitting: class data example
- I asked SAS to automatically find predictors of optimism in our class dataset. Here’s the resulting linear regression model:
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept 11.80175 2.98341 11.96067 15.65 0.0019
exercise -0.29106 0.09798 6.74569 8.83 0.0117
sleep -1.91592 0.39494 17.98818 23.53 0.0004
obama 1.73993 0.24352 39.01944 51.05 <.0001
Clinton -0.83128 0.17066 18.13489 23.73 0.0004
mathLove 0.45653 0.10668 13.99925 18.32 0.0011
Exercise, sleep, and high ratings for Clinton are negatively related to optimism (highly significant!) and high ratings for Obama and high love of math are positively related to optimism (highly significant!).
If something seems to good to be true…
Clinton, univariate:
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 5.43688 2.13476 2.55 0.0188
Clinton Clinton 1 0.24973 0.27111 0.92 0.3675
Sleep, Univariate:
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 8.30817 4.36984 1.90 0.0711
sleep sleep 1 -0.14484 0.65451 -0.22 0.8270
Exercise, Univariate:
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 6.65189 0.89153 7.46 <.0001
exercise exercise 1 0.19161 0.20709 0.93 0.3658
More univariate models…
Obama, Univariate:
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 0.82107 2.43137 0.34 0.7389
obama obama 1 0.87276 0.31973 2.73 0.0126
Love of Math, univariate:
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 3.70270 1.25302 2.96 0.0076
mathLove mathLove 1 0.59459 0.19225 3.09 0.0055
Compare with multivariate result; p<.0001
Compare with multivariate result; p=.0011
Overfitting
Pure noise variables still produce good R2 values if the model is overfitted. The distribution of R2 values from a series of simulated regression models containing only noise variables.
(Figure 1 from: Babyak, MA. What You See May Not Be What You Get: A Brief, Nontechnical Introduction to Overfitting in Regression-Type Models. Psychosomatic Medicine 66:411-421 (2004).)
Rule of thumb: You need at least 10 subjects for each additional predictor variable in the multivariate regression model.
Review of statistical tests
The following table gives the appropriate choice of a statistical test or measure of association for various types of data (outcome variables and predictor variables) by study design.
e.g., blood pressure= pounds + age + treatment (1/0)
Continuous outcome
Binary predictor
Continuous predictors
Types of variables to be analyzed
Statistical procedure
or measure of association
Predictor variable/s
Outcome variable
Cross-sectional/case-control studies
Categorical (>2 groups)
Continuous
ANOVA
Continuous
Continuous
Simple linear regression
Multivariate
(categorical and continuous)
Continuous
Multiple linear regression
Categorical
Categorical
Chi-square test (or Fisher’s exact)
Binary
Binary
Odds ratio, risk ratio
Multivariate
Binary
Logistic regression
Cohort Studies/Clinical Trials
Binary
Binary
Risk ratio
Categorical
Time-to-event
Kaplan-Meier/ log-rank test
Multivariate
Time-to-event
Cox-proportional hazards regression, hazard ratio
Binary (two groups)
Continuous
T-test
Binary
Ranks/ordinal
Wilcoxon rank-sum test
Categorical
Continuous
Repeated measures ANOVA
Multivariate
Continuous
Mixed models; GEE modeling
Alternative summary: statistics for various types of outcome data
| Outcome Variable | Are the observations independent or correlated? | Assumptions | |
| independent | correlated | ||
| Continuous (e.g. pain scale, cognitive function) | Ttest ANOVA Linear correlation Linear regression | Paired ttest Repeated-measures ANOVA Mixed models/GEE modeling | Outcome is normally distributed (important for small samples). Outcome and predictor have a linear relationship. |
| Binary or categorical (e.g. fracture yes/no) | Difference in proportions Relative risks Chi-square test Logistic regression | McNemar’s test Conditional logistic regression GEE modeling | Chi-square test assumes sufficient numbers in each cell (>=5) |
| Time-to-event (e.g. time to fracture) | Kaplan-Meier statistics Cox regression | n/a | Cox regression assumes proportional hazards between groups |
Continuous outcome (means); HRP 259/HRP 262
| Outcome Variable | Are the observations independent or correlated? | Alternatives if the normality assumption is violated (and small sample size): | |
| independent | correlated | ||
| Continuous (e.g. pain scale, cognitive function) | Ttest: compares means between two independent groups ANOVA: compares means between more than two independent groups Pearson’s correlation coefficient (linear correlation): shows linear correlation between two continuous variables Linear regression: multivariate regression technique used when the outcome is continuous; gives slopes | Paired ttest: compares means between two related groups (e.g., the same subjects before and after) Repeated-measures ANOVA: compares changes over time in the means of two or more groups (repeated measurements) Mixed models/GEE modeling: multivariate regression techniques to compare changes over time between two or more groups; gives rate of change over time | Non-parametric statistics Wilcoxon sign-rank test: non-parametric alternative to the paired ttest Wilcoxon sum-rank test (=Mann-Whitney U test): non-parametric alternative to the ttest Kruskal-Wallis test: non-parametric alternative to ANOVA Spearman rank correlation coefficient: non-parametric alternative to Pearson’s correlation coefficient |
Binary or categorical outcomes (proportions); HRP 259/HRP 261
| Outcome Variable | Are the observations correlated? | Alternative to the chi-square test if sparse cells: | |
| independent | correlated | ||
| Binary or categorical (e.g. fracture, yes/no) | Chi-square test: compares proportions between two or more groups Relative risks: odds ratios or risk ratios Logistic regression: multivariate technique used when outcome is binary; gives multivariate-adjusted odds ratios | McNemar’s chi-square test: compares binary outcome between correlated groups (e.g., before and after) Conditional logistic regression: multivariate regression technique for a binary outcome when groups are correlated (e.g., matched data) GEE modeling: multivariate regression technique for a binary outcome when groups are correlated (e.g., repeated measures) | Fisher’s exact test: compares proportions between independent groups when there are sparse data (some cells <5). McNemar’s exact test: compares proportions between correlated groups when there are sparse data (some cells <5). |
Time-to-event outcome (survival data); HRP 262
| Outcome Variable | Are the observation groups independent or correlated? | Modifications to Cox regression if proportional-hazards is violated: | |
| independent | correlated | ||
| Time-to-event (e.g., time to fracture) | Kaplan-Meier statistics: estimates survival functions for each group (usually displayed graphically); compares survival functions with log-rank test Cox regression: Multivariate technique for time-to-event data; gives multivariate-adjusted hazard ratios | n/a (already over time) | Time-dependent predictors or time-dependent hazard ratios (tricky!) |
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