1. Introduction to Biostatistical Analysis Using R Statistics course for first-year PhD students
Session 4
Lecture: Regression Analysis
Practical: multiple regression
Lecturer: Lorenzo Marini
DAFNAE,
University of Padova, Viale dell’Università 16, Legnaro, Padova.
Tel.:
Skype: lorenzo.marini

2. Statistical modelling: more than one parameter
Nature of the response variable
Generalized
Linear Models
NORMAL
POISSON,
BINOMIAL …
GLM
Session 3
General Linear Models
Nature of the explanatory variables
Categorical
Continuous
Categorical + continuous
ANOVA
Regression
ANCOVA
Session 3
Session 4

3. REGRESSION REGRESSION Linear methods Non-linear methods Simple linear
-One X
-Linear relation
Multiple linear
-2 or > Xi
- Linear relation
Non-linear
-One X
-Complex relation
Polynomial
-One X but more slopes
- Non-linear relation

4. LINEAR REGRESSION lm()
Regression analysis is a technique used for the modeling and analysis of numerical data consisting of values of a dependent variable (response variable) and of one or more independent continuous variables (explanatory variables)
Assumptions
Independence: The Y-values and the error terms must be independent of
each other.
Linearity between Y and X.
Normality: The populations of Y-values and the error terms are normally
distributed for each level of the predictor variable x
Homogeneity of variance: The populations of Y-values and the error terms have the same variance at each level of the predictor variable x.
(don’t test for normality or heteroscedasticity, check the residuals instead!)

5. We estimate one INTERCEPT
LINEAR REGRESSION lm()
AIMS
1.To describe the linear relationships between Y and Xi (EXPLANATORY APPROACH) and to quantify how much of the total variation in Y can be explained by the linear relationship with Xi.
2. To predict new values of Y from new values of Xi (PREDICTIVE APPROACH) Xi
predictors Y
response
Yi = α + βxi + εi
We estimate one INTERCEPT
and one or more SLOPES

6. SIMPLE LINEAR REGRESSION:
Simple LINEAR regression step by step:
I step:
-Check linearity [visualization with plot()]
II step:
-Estimate the parameters (one slope and one intercept)
III step:
-Check residuals (check the assumptions looking at the residuals: normality and homogeneity of variance)

7. SIMPLE LINEAR REGRESSION:
Normality
Do not test the normality over the whole y

8. SIMPLE LINEAR REGRESSION
MODEL
The model gives the fitted values
yi = α + βxi
SLOPE
β = Σ [(xi-xmean)(yi-ymean)]
Σ (xi-xmean)2
INTERCEPT
α = ymean- β*xmean
The model does not explained everything
RESIDUALS
Residuals= observed yi- fitted yi
Observed value

9. SIMPLE LINEAR REGRESSION
Least square regression explanation
library(animation)
###########################################
##Slope changing
# save the animation in HTML pages
ani.options(ani.height = 450, ani.width = 600, outdir = getwd(),
title = "Demonstration of Least Squares",
description = "We want to find an estimate for the slope
in 50 candidate slopes, so we just compute the RSS one by one. ")
ani.start()
par(mar = c(4, 4, 0.5, 0.1), mgp = c(2, 0.5, 0), tcl = -0.3)
least.squares()
ani.stop()
############################################
# Intercept changing
least.squares(ani.type = "i")

10. SIMPLE LINEAR REGRESSION
Parameter t testing
Hypothesis testing
Ho: β = 0 (There is no relation between X and Y)
H1: β ≠ 0
Parameter t testing (test the single parameter!)
We must measure the unreliability associated with each of the
estimated parameters (i.e. we need the standard errors)
SE(β) = [(residual SS/(n-2))/Σ(xi - xmean)]2
t = (β – 0) / SE(β)

11. SIMPLE LINEAR REGRESSION
If the model is significant (β ≠ 0)
How much variation is explained?
Measure of goodness-of-fit
Total SS = Σ(yobserved i- ymean)2
Model SS = Σ(yfitted i - ymean)2
Residual SS = Total SS - Model SS
R2 = Model SS /Total SS
It does not provide information
about the significance
Explained variation

12. SIMPLE LINEAR REGRESSION: example 1
If the model is significant, then model checking
1. Linearity between X and Y? ok
No patterns in the residuals vs. predictor plot

13. SIMPLE LINEAR REGRESSION: example 1
2. Normality of the residuals
Q-Q plot + Shapiro-Wilk test on the residuals ok
> shapiro.test(residuals)
Shapiro-Wilk normality test
data: residuals
W = , p-value = ok

14. SIMPLE LINEAR REGRESSION: example 1
3. Homoscedasticity
Call:
lm(formula = abs(residuals) ~ yfitted)
Coefficients:
Estimate SE t P
(Intercept)
yfitted ok

15. SIMPLE LINEAR REGRESSION: example 2
1. Linearity between X and Y? no no
yes
NO LINEARITY between X and Y

16. SIMPLE LINEAR REGRESSION: example 2
2. Normality of the residuals
Q-Q plot + Shapiro-Wilk test on the residuals no
> shapiro.test(residuals)
Shapiro-Wilk normality test
data: residuals
W = , p-value = no

17. SIMPLE LINEAR REGRESSION: example 2
3. Homoscedasticity NO
YES

18. SIMPLE LINEAR REGRESSION: example 2
How to deal with non-linearity and non-normality situations?
Transformation of the data
-Box-cox transformation (power transformation of the response)
-Square-root transformation
-Log transformation
-Arcsin transformation
Polynomial regression
Regression with multiple terms (linear, quadratic, and cubic)
Y= a + b1X + b2X2 + b3X3 + error X is one variable!!!

19. POLYNOMIAL REGRESSION: one X, n parameters
Hierarchy in the testing (always test the highest)!!!!
n.s.
n.s.
n.s.
No relation X
X + X2 + X3
X + X2
P<0.01
P<0.01
P<0.01
Stop
Stop
Stop
NB Do not delete lower terms even if non-significant

20. MULTIPLE LINEAR REGRESSION: more than one x
Multiple regression
Regression with two or more variables
Y= a + b1X1 + b2X2 +… + biXi
Assumptions
Same assumptions as in the simple linear regression!!!
The Multiple Regression Model
There are important issues involved in carrying out a multiple regression:
• which explanatory variables to include (VARIABLE SELECTION);
• NON-LINEARITY in the response to the explanatory variables;
• INTERACTIONS between explanatory variables;
• correlation between explanatory variables (COLLINEARITY);
• RELATIVE IMPORTANCE of variables

21. MULTIPLE LINEAR REGRESSION: more than one x
Multiple regression MODEL
Regression with two or more variables
Y = a+ b1X1+ b2X2+…+ biXi
Each slope (bi) is a partial regression coefficient:
bi are the most important parameters of the multiple regression model. They measure the expected change in the dependent variable associated with a one unit change in an independent variable holding the other independent variables constant. This interpretation of partial regression coefficients is very important because independent variables are often correlated with one another.

22. MULTIPLE LINEAR REGRESSION: more than one x
Multiple regression MODEL EXPANDED
We can add polynomial terms and interactions
Y= a + linear terms + quadratic & cubic terms+ interactions
QUADRATIC AND CUBIC TERMS account for NON-LINEARITY
INTERACTIONS account for non-independent effects of the factors

23. MULTIPLE LINEAR REGRESSION:
Multiple regression step by step:
I step:
-Check collinearity (visualization with pairs() and correlation)
-Check linearity
II step:
-Variable selection and model building (different procedures to select the significant variables)
III step:
-Check residuals (check the assumptions looking at the residuals: normality and homogeneity of variance)

24. MULTIPLE LINEAR REGRESSION: I STEP
-Check collinearity
-Check linearity
Let’s begin with an example from air pollution studies. How is ozone
concentration related to wind speed, air temperature and the intensity
of solar radiation?

25. II STEP: MODEL BUILDING
MULTIPLE LINEAR REGRESSION: II STEP
II STEP: MODEL BUILDING
Start with a complex model with interactions and quadratic
and cubic terms
Model simplification
Minimum Adequate Model
How to carry out a model simplification in multiple regression
1. Remove non-significant interaction terms.
2. Remove non-significant quadratic or other non-linear terms.
3. Remove non-significant explanatory variables.
4. Amalgamate explanatory variables that have similar parameter values.

26. MULTIPLE LINEAR REGRESSION: II STEP
Start with the most complicate model (it is one approach)
model1<lm( ozone ~ temp*wind*rad+I(rad2)+I(temp2+I(wind2))
Estimate
Std.Error t
Pr(>t)
(Intercept)
5.7E+02
2.1E+02
2.74
0.01 **
temp
-1.1E+01
4.3E+00
-2.50 *
wind
-3.2E+01
1.2E+01
-2.76
rad
-3.1E-01
5.6E-01
-0.56
0.58
I(rad^2)
-3.6E-04
2.6E-04
-1.41
0.16
I(temp^2)
5.8E-02
2.4E-02
2.44
0.02
I(wind^2)
6.1E-01
1.5E-01
4.16
0.00
***
temp:wind
2.4E-01
1.4E-01
1.74
0.09
temp:rad
8.4E-03
7.5E-03
1.12
0.27
wind:rad
2.1E-02
4.9E-02
0.42
0.68
temp:wind:rad
-4.3E-04
6.6E-04
-0.66
0.51
!!!!!!
We cannot delete these terms
!!!!!!!
Delete only the highest interaction temp:wind:rad

27. MULTIPLE LINEAR REGRESSION: II STEP
Manual model simplification
(It is one of the many philosophies)
Deletion the non-significant terms one by one:
Hierarchy in the deletion:
1. Highest interactions
2. Cubic terms
3. Quadratic terms
4. Linear terms
COMPLEX
Deletion
At each deletion test:
Is the fit of a
simpler model worse?
SIMPLE
IMPORTANT!!!
If you have quadratic and cubic terms significant you cannot
delete the linear or the quadratic term even if they are not significant
If you have an interaction significant you cannot
delete the linear terms even if they are not significant

28. MULTIPLE LINEAR REGRESSION: III STEP
III STEP: we must check the assumptions NO NO
Variance tends to increase with y
Non-normal errors
We can transform the data (e.g. Log-transformation of y)
model<lm( log(ozone) ~ temp + wind + rad + I(wind2))

29. MULTIPLE LINEAR REGRESSION: more than one x
The log-transformation has improved our model but maybe there is an outlier

30. PARTIAL REGRESSION:
With partial regression we can remove the effect of one or more variables (covariates) and test a further factor which becomes independent from the covariates
WHEN?
Would like to hold third variables constant, but cannot manipulate.
Can use statistical control.
HOW?
Statistical control is based on residuals. If we regress Y on X1 and take residuals of Y, this part of Y will be uncorrelated with X1, so anything Y residuals correlate with will not be explained by X1.

31. PARTIAL REGRESSION: VARIATION PARTITIONING
Relative importance of groups of explanatory variables
R2= 76% (TOTAL EXPLAINED VARIATION)
What is space and what is environment?
Total variation
Latitude (km)
Space ∩
Envir.
Unexpl.
Space
Environment
Longitude (km)
Explained variation
Full.model<lm(species ~ environment i + space i)
Site
Response variable: orthopteran species richness
Explanatory variable: SPACE (latitude + longitude) +
ENVIRONMENT (temperature + land-cover heterogeneity)

32. VARIATION PARTITIONING: varpart(vegan)
Full.model<lm(SPECIES ~ temp + het + lat + long)
Unexpl.
Space
Environment
TVE=76%
Env.model<lm(SPECIES ~ temp + het)
env.residuals
Unexpl.
Space
Environment
Pure.Space.model<lm(ENV.RESIDUALS ~ lat + long)
VE=15%
Unexpl.
Space
Environment
Space.model<lm(SPECIES ~ lat + long)
space.residuals
Unexpl.
Space
Environment
Pure.env.model<lm(SPACE.RESIDUALS ~ tem + het)
VE=40%
Unexpl.
Space
Environment

33. NON-LINEAR REGRESSION: nls()
Sometimes we have a mechanistic model for the relationship between y and x, and we want to estimate the parameters and standard errors of the parameters of a specific non-linear equation from data.
We must SPECIFY the exact nature of the function as part of the model formula when we use non-linear modelling
In place of lm() we write nls() (this stands for ‘non-linear least squares’). Then, instead of y~x+I(x2)+I(x3) (polynomial), we write the y~function to spell out the precise nonlinear model we want R to fit to the data.

34. Model weights and model average
NON-LINEAR REGRESSION: step by step
1. Plot y against x
Alternative approach
2. Get an idea of the family of functions that you can fit
Multimodel inference
(minimum deviance +
minimum number of parameters)
3. Start fitting the different models
AIC = scaled deviance +2k
4. Specify initial guesses for the values of the parameters
k= parameter number + 1
Compare GROUPS of model at a time
5. [Get the MAM for each by model simplification]
6. Check the residuals
Model weights and model average
[see Burnham & Anderson, 2002]
7. Compare PAIRS of models and choose the best

35. Exponential functions
nls(): examples of function families
Asymptotic functions
S-shaped functions
Humped functions
Exponential functions

36. Asymptotic exponential
nls(): Look at the data
Asymptotic functions ?
S-shaped functions
Humped functions
Exponential functions
Asymptotic exponential
Understand the role of the parameters a, b, and c
Using the data plot work out sensible starting values. It always helps in cases like this to work out the equation’s at the limits – i.e. find the values of y when x=0 and when x=

37. nls(): Look at the data Fit the model
1. Estimate of a, b, and c (iterative)
2. Extract the fitted values (yi)
3. Check graphically the curve fitting
Can we try another function from the same family?
Model choice is always an important issue in curve fitting
(particularly for prediction)
Different behavior at the limits!
Think about your biological system not just residual deviance!

38. nls(): Look at the data Michaelis–Menten Fit a second model
1. Extract the fitted values (yi)
2. Check graphically the curve fitting
You can see that the asymptotic exponential (solid line) tends to get to its asymptote first, and that the Michaelis–Menten (dotted line) continues to increase. Model choice, therefore would be enormously important if you intended to use the model for prediction to ages much greater than 50 months.

39. Application of regression: prediction
Regression models for prediction
A model can be used to predict values of y in space or in time
knowing new xi values
Spatial extent + data range
YES NO
Before using a model for prediction it has to
be VALIDATED!!!
2 APPROACHES

40. VALIDATION
1. In data-rich situation, set aside validation (use one part of data set to fit model, second part for assessing prediction error of final selected model).
Residual=Prediction error
Real y
Model fit
Predicted
2. If data scarce, must resort to “artificially produced” validation sets
Cross-validation
Bootstrap

41. Cross-validation estimate of prediction error is average of these
K-FOLD CROSS-VALIDATION
Split randomly the data in K groups with roughly the same size
Take turns using one group as test set and the other k-I as training set for fitting the model 1 2 3 4 5
1. Prediction error1
Test
Train
Train
Train
Train
Train
Train
Test
Train
Train
2. Prediction error2
3. Prediction error3
Train
Train
Train
Test
Train
4. Prediction error4
Train
Test
Train
Train
Train
5. Prediction error5
Train
Train
Train
Train
Test
Cross-validation estimate of prediction error is average of these

42. 2. For each bootstrap sample, compute the prediction error
1. Generate a large number (n= ) of bootstrap samples …
n=10000
2. For each bootstrap sample, compute the prediction error
Error1
Error2
Error3 …
Errorn
3. The mean of these estimates is the bootstrap estimate of prediction error

43. Application of regression: prediction
1. Do not use your model for prediction without carrying out a validation
If you can, use an independent data set for validating the model
If you cannot, use at least bootstrap or cross-validation
2. Never extrapolate