Presentation on theme: "Getting More out of Multiple Regression Darren Campbell, PhD."— Presentation transcript:
Getting More out of Multiple Regression Darren Campbell, PhD
Overview View on Teaching Statistics When to Apply How to Use & How to Interpret
Multiple Regression Techniques 1. Centring removing /group difference confounds 2. Centring interpret continuous interactions 3. Spline functions – Piecemeal Polynomials Estimate separate slopes each angle of the regression polynomial
Perks of Multiple Regression 1. Realistic many influences Behaviour 2. Control over confounds 3. Test for relative importance 4. Identify interactions
Why Not Use ANOVAs? Not realistic: Many behaviours / constructs are continuous e.g., intelligence, personality Loss of statistical power - categories scores assumed to be the same + error mixing systematic patterns into the error term
What is Centring? Simple re-scaling of raw scores Raw Score minus Some Constant value x1 – – 5.1 = – 5.1 = -1.1 x2 – – 29.4 = = 5.6
A Simple Case for Centring Babies: Cry & Fuss – parent report diary measures Fail about - limb movement Are these 2 infant behaviours related? Emotional Responses & Emotion Regulation
A Simple Case for Centring AgeMoves / HrCrying Hrs/Day 6 week olds month olds Full Sample Are these 2 infant behaviours related?
6 Week-Olds r = +.47 some infants cry more & move more others cry less & move less
6 Month-Olds r = +.38 some infants cry more & move more others cry less & move less What if we combine the two groups?
Full sample r = Do we get a significant corr? If so, what kind?
What happened with the Correlations? 6 Week-olds: r = Month-Olds: r = Week & 6 Month-olds: r = -0.22
Correlations = Grand Mean Centring 1) Mean Deviations for each variable: X & Y 2) Rank Order Mean Deviations 3) Correlate 2 rank orders of X & Y
The Disappearing Correlation Explained Grand Mean Centring lead to all the older infants being classified as high movers young infants low movers Young high criers & high movers -> high criers & low movers Large Group differences in movement altered the detection of within-group r’s What should we do?
Solution: Create Group Mean Deviations Re-scale raw scores Raw – Group Mean 6 week-olds: xs – month-olds: xs – 29.4
Solution: Create Group Mean Deviations CryingRaw AL Group Means Group Centred AL
Group Centred Scores Group mean data r =.41 - full sample Mulitple Regression could also work on uncentred variables Crying = Group + Uncentred AL Not a Group x AL interaction – the relation is the same for both groups
Centring so far 1. Centring is Magic 2. Different types of centring Depending on the number used to re-scale the data Grand mean – Pearson Correlations Group Means – Infant Limb Movements
Regression Interactions Centring Great for Interpreting Interactions trickier than for ANOVAs do not have pre-defined levels or groups based on 2+ continuous vars
Multiple Regression - the Basics The Basic Equation: Y = a + b1*X 1 + b2*X 2 + b3*X 3 + e Outcome = Intercept + Beta1 * predictor1 + B2 * pred2 + B3 * pred3 + Error a = expected mean response of y betas: every 1 unit change in X you get a beta sized change in Y
Regression Interactions Centring Reducing multicollinearity interaction predictor = x1 * x2 x1 & x2 numbers near 0 stay near 0 and high x1 & x2 numbers get really high interaction term is highly correlated with original x1 & x2 variables Centring makes each predictor: x1 & x2 have more moderate numbers above and below zero positive and negative numbers Reduces the multiplicative exaggeration between x1 & x2 and the interaction product x1*x2
Centring to reduce Multicollinearity
Regression Y = a + b1*X 1 + b2*X 2 + b3*X 1 *X 3 + e How does X2 relate to Y at different levels of X1? How does predictor 2 (shyness) relate to the outcome (social interactions) at different stress levels (X1)?
Uncentred DataCentred Data X1 = 26.2 (14.5)X1 = 0.0 (14.5) X2 = 24.8 (27.6)X2 = 0.0 (27.6) x1x2x12yx1cx2cx12cy x **0.65**0.14** x1c ** * x **0.28** x2c **0.28** x ** x12c ** Correlation Matrix: ** p =.01 * p =.05
Regression Equation Results No Interaction: Y = b0 + b1 * X1 + b2 * X2 Uncentred: Y = – 4 X X2 ** Centred: Y = – 4 X X2 **
Regression Equation Results Interaction Term Included: Y = b0 + b1 * X1 + b2 * X2 + b3 * X1*X2 Uncentred: Y = 1733 – 19.1 X1 – 31.7 X2 ** X1*X2 Centred: Y = X X X1*X2
But what does it mean… How does X2 relate to Y at different levels of X1? How does predictor 2 (shyness) relate to the outcome (social interactions) at different stress levels (X1)?
Post Hocs Y = b0 + b1 * X1 + b2 * X2 + b3 * X1*X2 Y = ( b1 * X1 + b0 ) + ( b2 + b3 * X1 ) * X2 -1 SD below X1 Mean& + 1SD above X1 Mean X - ( )X X
Scatterplots: Moving the Y Axis
-1 SD Below X1 Mean Y = X X X1*X2 t (1,196) = -1.40, p =.16 Centred: Y = X X X1*X2 t (1,196) = 0.12, p = SD Above X1 Mean Y = X X2 ** X1*X2 t (1,196) = 3.66, p =.001
Regression Interaction Example Predicting inhibitory ability with motor activity & age simon says like games 4 to 6 yr-olds & physical movement Move by Age interaction F (1, 81) = 5.9, p <.02 Young (-1.5SD): move beta sig + Inhibition Middle (Mean) : move beta p =.10 ~ Inhibition Older (+1.5SD): move beta n.s. inhibition
Polynomials, Centring, & Spline Functions Polynomial relations: quadratic, cubic, etc Y = a + b1*X 1 - b2*X 1 *X 1 + e
Curvilinear Pattern Assume a symmetric pattern – X 2 But, it may not be... Perceived Control (Y) slowly increases & then declines rapidly in old age
This Brings us to Spline Functions Split up predictor X 2+ variables X Low & X High X Low = X – (-5) & set values at the next change point to zero Ditto for X High Re-run Y = a + b1*X Low - b2*X High + e
Perks of Spline Functions Estimate slope anywhere along the range Can be sig on one part - n.s. on another Steeper or shallower
Questions Alpha control procedures for spline functions – Could be argue that you are describing the pattern already identified? – Conservatively, you could apply an alpha control procedure. I like the False Discovery Rate procedures. – Replication is preferred, but not always possible.
Alpha Control Aside The source of Type 1 errors is typically poorly described. Typical: If enough probability tests are run, the probability will increase to the point where something becomes significant just by chance. – But, probability is linked to the representativeness of your data and type 1 error is a proxy for the likelihood of the representativeness of your data. My View: The real source of Type 1 errors is that if you – divide up the data into enough subgroupings – eventually one of those subgroupings will differ because it is misrepresentative of reality.
Standardized vs Centred Centred is x – x M Standardized (x – x M )/ SDx – Makes variability for each predictor = 1 – Standardized Beta = raw b * SDx / SDy – Similar to centring but different metric needs to be adjusted for interaction terms To get comparable results with interaction term – Standardization should be applied to X1 and X2 prior to the X1*X2 estimate then use “raw” coefficients
Centring and Spline Functions Relatively simple procedures Old dogs in the Statistic World but new tricks for many That’s All Folks!