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Simple Line Regression
regression
performed to test if there is a significant prediction of a DV by an IV (predictor)
DV is interval/ratio
unidirectional

Assumptions of Regression
regression
Normality of the errors
No Outliers
Linearity
Homoscedasticity
Independence of subjects

Regression Assumptions: Normality of the errors
regression
the distribution of erros is normal
assumption tested using ShapiroWilk

Regression Assumptions: No Outliers
regression
no extreme scors among errors.
assumption tested by examining histograms and QQ plots

Regression Assumptions: Linearity
regression
the relationship between DV and IV is linear
 tested by examing scatterplots of:
 1. DV vs IV
 (and/or)
 2. errors vs predicted scores
can be objectively tested using Box Cox transformation

Regression Assumptions: Homoscedasticity
regression
variability of DV is same across the levels of an IV
 tested by examining scatterplots of:
 1. DV vs IV
 and/or
 2.errors vs predicted scores
can be objectively tested using BrueschPagan or White's tests

Regression Assumptions: Independence of Subjects
regression
design consideration
cannot be tested using statistics
one subject's scores cannot be influenced by another subject

Regression Model
regression
 model in statistics that defines a relationship of variable(s) to another variable(s)
 how they are related to each other
 B_{0 }is yintercept
 B_{1} is the regression coefficient (slope)
 E_{i} is the error (residual)

Interpretations
regression
 b_{0 }is the value of DV predicted, Yhat_{i}, when the IV is 0, x_{i} = 0.
 Most of the time b_{0 }is not of any interest
 b_{1 }measures the amount of change in DV predicted, for a single unit increase in IV

Ordinary Least Squares
regression
estimates the parameters by minimizing the sum of square errors
mispredictions should be as small as possible

Errors
regression
difference between the DV and predicted DV
 some are positive while others are negative
 below line due to overestimation of scores
 above line due to underestimation of scores
Distance between score and line is the error

Standard Error of the Estimate
regression
standard deviation of the errors
difference between the observed and predicted DV
"on average, how much each subject is mispredicted by"

Research question: is there a significant prediction of a DV by an IV?
regression
ttest for b_{1}
df = N  2

Research question: is the yintercept significantly different than 0?
regression
 ttest for b_{0}
 df = N  2

Effect Size
regression
 measure of effect that is often used in simple linear regression is the coefficient of determination, R^{2}
 which measures the proportion of the variability of a DV explained by an IV
 ranges from 0 to 1
 used only for descriptive purposes
 descripes sample not population

Adjusted Rsquare
regression
used to make inferences about population
adjusts for sample size (and the number of IVs)

Beta
regression
_{(spelled out for standardized regression coefficient)}
measures the amount of change in standard unit (i.e. 1 SD) of DV predicted for a one standard unit increase in the DV
same as r between a DV and IV

