Linear Regression

correlation =

relationship

regression =

prediction and explanation

How do we use predictions in PT

setting realistic goals

how to allocate treatment and select interventions

develop plan of care

predict prognosis

 What is the purpose of the coefficient of determination (r²)

indicates the proportion of variance that is shared by 2 variables, or that portion of variance that can be explained by knowing the value of X

What is regression

used to predict and to explain variance in a set of data

example of regression

What characteristics predict full functional recovery after a CVA

What are good for prognosis clinical questions

regression

What are the 3 types of regression

linear

multiple linear regression

logistic 

describe linear regression

consistent relationship between 2 variables

What are the names for X in a linear regression

independent, predictor

What are the names for Y in a linear regression

dependent, predicted (criterion)

What can occur if the correlation is perfect in a linear regression

can use any value of X to state the value of Y

What is the regression line equation

Y hat = mX + b

Y hat = a + bX

What does the regression line mean

predicted value of Y is the values of X times the slope of the regression line plus the Y-axis intercept

What is the regression constant

a or b, y-intercept

What is the regression coefficient

b or m, slope

What are residuals

the distance between the data point and the line (the actual value minus the predicted value)

What should the sum of all the residuals be equal to

0

the sum of the squared residuals will be smallest when..

the regression line is best (best fit line/least squares method)

correlation coefficient gives…

strength of relationship 

coefficient of determination indicates…

the accuracy of the prediction

What is the analysis mean if the residuals are horizontal

no problem, assumptions have been met

What is the analysis mean if the residuals are fanned out

increased error with larger values, assumptions are a not met 

What is the analysis mean if the residuals are curved

no longer valid for linear model

What does a correlation of ±1, what is the regression ling indicative of

a strong basis for prediction

our error in prediction increases as r gets…

smaller

describe the coefficient of determination (r²)

a measure of how much variance in Y can be explained by knowing X

What is the scale for r²

0-1

What do correlation coefficients indicate

strength of relationship

What does the coefficient of determination indicate

accuracy of prediction

define standard error of the estimate (SEE)

variance of errors on either side of the regression lines

What are residuals

the distance between outliers and the mean line

What can we use to determine SEE

normal distribution

confidence intervals

How does SEE affect our prediction

increased variance → increased dispersion around the line → decreased accuracy of prediction

SEE relationship with CI

the larger the SEE, the wider the CI

How do we know that the observed relationship did not occur by chance

correlation coefficient

ANOVA of regression

T test of slope

What do we want our F value to be

the larger the better

if the relationship is not significant, does that mean that they are not related?

not necessarily (may not be a linear relationship)

What is a 1st order regression

linear

What is a 2nd order regression

quadratic (U shape)

What is a 3rd order regression

cubic (sin wave)

What is a 4th order regression

curve changes 3 times

What type of relationship do regressions assume

general linear model

How do we know when to use linear vs polynomial regression models

look at the scatterplot

What is a multiple linear regression model

takes onto consideration multiple variables affecting a outcome

procedure for the multiple linear regression model

add another IV

what % of variation is explained by the model (both IVs)

what % of variation is explained by both IVs separately

repeat until 100% of variation in the DV is explained

utility of multivariate regression models

• permit analysis of IVs on 1 DV (both continuous and categorical variables)

• assists in determining which IVs are most important

• DV must be continuous

• helpful in prognosis

Multiple regression equation

Y=a+bX (1)+ bx (2)

define multicollinearity

when IVs are correlated with each other

What is the issue with collinearity

presents problem for interpretation of beta weights

How is collinearity measured

tolerance level

variance inflation factor (VIF)

describe the scale for tolerance level

0-1 (1=unique)

describe the scale for VIF level

1+ (1=unique)

effect size for regression

R²

interpretation for a small R²

.02

interpretation for a medium R²

.13

interpretation for a large R²

.26

describe stepwise multiple regression

uses specific stat criteria to retain or eliminate variables to maximize prediction accuracy

purpose of stepwise multiple regression

allows us to determine which combo of factors are the most meaningful

describe a backwards stepwise regression

removing one variable at a time to determine which one causes an effect

stepwise regression steps

1. IVs are correlated with DV

2. the highest ranking r is entered into model (BMI)

3. remaining IVs are examined for “partial correlation” (BMI removed)

4. repeat

how do we know when to stop adding variables

When all variables are accounted for, or the addition of variables isn’t making a significant improvement in predictions

What are we looking for with Beta

the largest variable (contributes the most to outcome)