1
Q
Regression is?
A
- More Fiddly than other methods
- Has more assumptions
2
Q
Why do Linear Regression
A
- Not looking at differences
- Looking at relationships
- Regression goes further than correlation - Allows us to make predictions
- Produces a model that allows for sophisticated exploration of relationships in variables
3
Q
In Second Year Stats
A
- Looked at relationships - Correlation
- Differences Between Groups and Within-Groups
- Used t-tests and aNOVAs
- Variation in Dependent Variable
4
Q
Correlation
A
Allows us to estimate direction and strength of a linear relationship
5
Q
Why do Linear Regression
A
- How well will a set of variables predict an outcome?
- Which variable in a set of variables is the best predictor of an outcome?
- Does a particular predictor variable predict an outcome if another variable is controlled for?
6
Q
Predictor Variable
A
Same as Independent Variable in Regression
7
Q
Outcome Variable
A
The same as the Dependent Variable in Regression
8
Q
What is a Model?
A
- An approximation to the actual data
- simple summary of data
- Makes data easier to interpret, communicate
- Allows us to predict data
9
Q
What is a Regression Model
A
Mathematically Describes the linear relationship
* Y = Beta X + C
* Y = Predicted valuus of the DV
* Beta = The slope of the line
* X = Scores on the Predictor (IV)
* C = The Intercept
10
Q
The Intercept
A
- Point where the function crosses the y-axis.
- Sometimes Regression model only becomes significant when we remove the intercept, and the regression line reduces to
- Y = b(X) + error
11
Q
Standardized beta (β)
A
- Compares the strength of the effect of each IV to the DV
- The higher the absolute value of the beta coefficient, the stronger the effect
11
Q
How Does Regression Work?
A
- Linear combination of another variable don’t always have to be continuous
- Can have a combination of variables
- Need to find the Line of Best Fit
12
Q
Line of Best Fit
A
- Many lines produced with Regression Formula
- How do we know what line is best?
- Mimimises the difference between observed values and data predicted by the line
- This is called error
- In regression also called residuals
* Y = b(X) + C + error
13
Q
N (Cases): k (Predictors) Ratio
A
- Assumption about sample size
- Need a certain number of participants to trust validity
- Simple Linear Regression Assumption
- Number of Cases multiplied by Predictors
- The more Predictors we have the more cases we need for the study
14
Q
Checking Linearity
A
- checking for Linearity requires scatter plots
- Need Scattepots between each DV & IV
- Looking for Non-Linear evidence
15
Q
Check for Normality
A
- Kolmogorov-Smirnov/Shapiro-Wilks: p > .05
- Skewness & Kurtosis: z score is < ±1.96 then it is normal
- Histogram follows a bell curve.
- Detrended Q-Q Plots: Equal amounts of dots above and below the line.
- Normal QQ Plots: Normal if dots hugging the line.
16
Q
Check for Univariate Outliers
Week 12 Part 2 - 10:00
A
- Identified on Box & Whisker Plots
- Dots indicate outliers
- Asterisk indicates extreme cases
- Number tells you which case is the issue
17
Q
Reason Univariate Outliers are Problematic
A
- Regression Analysis gives formula for a straight line
- A data point that stands outside other data points can change the slope of your straight line
- This makes the line a poor predictor of the value of other data points
17
Q
How to deal with Outliers
A
- Check if Outlier is a data entry error and fix it
- Check if outlier is from different population - Justifies removing their data
- Separate outliers and run different analysis
- Run Analysis with and without outliers and report both models
- Winsorization - Change values so they’re not Outliers anymore
- Use transformations or Bootstrapping
18
Q
Winsorization
A
- Change the score of outlier to value of 5th percentile for minimum values
- Change the score of outlier to value of 95th percentile for maximum values
- Slightly problematic because it changes the data
- But this retains extremeness without removing outlier data
19
Q
Bootstrapping
A
- Uses transformations to deal with outliers
- Creates samples from your sample
- Uses your Mean and Standard Deviation to create another data set
- does this repeatedly
- This creates a large data set where extreme values are more normal
19
Q
Homoscedasticity
A
- Means Scame Scatter or Same Variance
- Residuals are equal for all scores on the Outcome Variable
19
Q
Check for Normality, Linearity and Homoscedasticity
A
- We need the residuals to behave in a certain way
- Residuals are the difference between predicted scores and outcome variable
- SPSS generates a Histogram Q-Q Plots
20
Q
Dealing with Heteroscedasticity
A
- Check graph in SPSS
- Check the data for any patterns
- If dots are scattered randomly then we are all good
21
If Regression Assumptions are violated
* Check Normality of predictors, if you fix these heteroscedasitcity can dissapear
* Use a transformation on the Outcome Variable
* Consider using a different method like Weighted Least Squares Regression
* Use some kind of Non-Linear Regression
22
Null Hypothesis for Regression
* Slope of Regression line will be equal to 0
* Beta = 0
23
Alternative Hypothesis
* Slope of the Regression line will not be Zero
* Beta Not= 0
24
Running Linear Regression
**1. Analyse
2. Regression
3. Linear**
4. Move your DV into the dependent box.
5. Move your independent variable into the independent box.
**6. Ok**
25
Linear Regression - *R* value
* Same as Pearson's Correlation - *p* value
* Tells us strength and direction of relationship
26
Linear Regression *R* Square Value
* Tells us amount of variance in DV explained by IV
* Proportion of Variance that can be explained by the variable
* 23% of variability in grades explained by attendance in this example
* Known to overestimate the explained variance
27
Linear Regression - *R* Square Adjusted
* *R* needs to be adjusted to be smaller than *R* squared
* Corrects bias of overestimated explained variance
* Useful as Goodness of Fit Statistic
28
Goodness of Fit Statistic
* Determines how well sample data fits a distribution from a normal population
* Determines if a sample is skewed or normal in the actual population
29
Regression ANOVA
* Uses the df, *F* value and the *p* value
* Compares error rate with line of best fit and the error rate of the baseline model of 0
* ANOVA is significant if it is "better" than the baseline
30
Unstandardised Coefficient
* The Slope of the Regression Equation
* Amount of change in a Dependent Variable due to a change of an Independent Variable
* This is the Beta coefficient
e.g. each unit of attendance is associated with 1.88 unit of increase in grades
31
Coefficient t-tests
* Check if IV is a significant predictor of the DV
* Become more relevant when we start adding more predictors
32
Standardised Coefficients
* A measure of the effect size
* Useful for multiple Regression
* Important when we have more than one Predictor
* Predictors often measured in different scales
* e.g, IQ Points, Classes attended, additional study time
33
Dealing with Multiple Predictors
* Most commonly found in Research Projects
* Allows us to predict the outome variable from more than one predictor
* Answers how well does combination predictors predict the outcome
**Y = b1(X1) + b2(X2) + C + error**
34
Univariate OUtliers
Outlier on one variable
35
Multivariate Outlier
Outlier on a combination of variables
36
Assumptions with Regression
* Normality
* Univariate Outliers
* Multivariate Outliers
* Multicollinearity
* Normality, Linearity & Homosedasticity of residuals
37
Multicollinearity
* Two or more IV's highly correlated in regression
* IV can be predicted from another IV in a regression model.
38
How to check for Multivariate Outliers
Mahalonobis Distance
39
Mahalanobis Distance
* largest value should not be greater than the critical 𝜒2 value for df = k at 𝛼= .001.
* Where k = the number of predictors.
* Use same table as Cook's Distance
* For simplicity use table below:
40
Cooks Distance
* Tells you if there are cases that influence the regression line
* Use same table as Mahalanobis Distance
* rule of thumb is if Cook’s **D is > 1** you have influential cases.
* Dealt with the same way as Univariate Oultiers
41
Check for Multicollinearity
* Pearson's Correlations between IV
* if *i* > .85 then there is multicollinearity
* **Tolerance:** Values < .1 are multicollinear; < .2 warrant a closer look
* **VIF:** Values > 10 are clearly Multicollinear; > 5 warrant a closer look
* If you find a problem then remove the offending variable
* If they are so closely related then they are basically the same thing. treat as one variable.
42
Check for Multivariate Outliers
* Use Residual Statistics Table
* First, we find the critical 𝜒2 for a model with 4 predictors: 𝜒2 = 18.467 - Check the Mahalanobis Distance Table
* Use Mahal. Distance Maximum (13.803 here)
* 13.803 < 18.467 Therefore there are no multivariate outliers.
* Cooks D is < 1 so there are no influential cases
43
Interpreting Multiple Regression
* Use the Variables Entered/Removed Table - Tells you how many predictors are in the model (4)
* Then Model Summary Table
* *R* is not just Person's *R* anymore
* It is correlation between actual scores and predictions in the regression equation
* *R square* = Proportion of variance in DV Accounted for y combined predictors
* Again *R square* Adjusted is a corrected version of *R square* that accounts for the positive bias.
44
Interpreting Multiple Regression ANOVA
* Now tests the comBination of predictors
* A significant predictor of GHQ
* The table has the df, the F value, and the p-value.
45
Interpreting Multiple Regression Coefficients
* Unstandardized coefficient is the slope of the regression
* Shows each unit increase in one of the independent variables is associated with a b unit increase in GHQ
* All other IVs are kept constant
* Beta values = Standardised regression coefficients
* Allow direct comparison of regression coefficients.
* Displayed in units of standard deviation.
46
Interpreting Multiple Regression Standardised Coefficients
* t-values and p-values test the significance of the unique contribution of each predictor
* Changes depending on predictors included in the model.
47
Multiple Regression Tolerance & VIF
* **Tolerance:** values < .1 are multicollinear; < .2 warrant closer inspection.
* **VIF:** values > 10 are clearly multicollinear; > 5 warrant closer inspection.
48
Remove Non-Significant Predictors
* If you have a predictor that is not reflecting anything it makes the model worse
* This changes the numbers slightly
* Only have significant predictors in the model
49
Applied look at Regression Equation
* Our general form for the regression is:
Y = b1(X1) + b2(X2) + b3(X3) + C + error
* And if we take this equation and substitute in our variables we get:
GHQ = b1(neuroticism) + b2(state-anxiety) + b3(trait-anxiety) + C + error
GHQ = .555(Neuroticism) + .318(state-anxiety) + .471(trait-anxiety) + 13.552 + error
50
What is the value for *R*?
* Correlation between theDV & IV
* Value greater than 0.4 is taken for further analysis.
51
What does *R* tell us?
The strength & direction of the relationship
52
What does the value of *R*2 Adjusted tell the researcher
* Tells you the percentage of variation explained by only the independent variables that actually affect the dependent variable.
53
What does the value of *R*2 Adjusted tell the researcher
* Tells you the percentage of variation explained by only the independent variables that actually affect the dependent variable.
54
What does the value of *R*2 Adjusted tell the researcher
* Tells you the percentage of variation explained by only the independent variables that actually affect the dependent variable.
55
What does the value of *R*2 Adjusted tell the researcher
* Tells you the percentage of variation explained by only the independent variables
* Those that actually affect the dependent variable.
PSYC3102 - HSA3
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