Statistical Control• Regression & research designs• Statistical Control is about – underlying causation– under-specification– measurement validity– “Correcting” the bivariate relationship • Regression & Residual formulas• Correlations among y, x, y’, & residuals• Control for Design & Measurement problems• “Controlling” proxy variables • Limitations & advantages of statistical controlExperimental “vs.” Statistical ControlIn the research design course (Psyc 941) we emphasized “experimental control”• control produced by the actions of the researcher• randomizing, balancing, holding constant, IV manipulation,etc.Another important form of control is “Statistical control”• control produced by applying the proper statistical analyses• usually used as an attempted substitute for “holding constant”Statistical control is generally considered inferior to experimental control, and often involves assumptions or changes in the research question that are “iffy”. But, the results from statistical control are often the first step toward an investment in acquiring experimental control over a “potential confounding variable”.Originally, linear regression was developed for use with agricultural data collected as part of true experiments.• The IV was usually some quantitative manipulation, such as amount of fertilizer, water, or seed and the DV was usually some assessment of “yield”.• This was a manipulated IV, so the value of the predictor was known with great precision (“no error”)• This was a true experiment, so causal interpretations were made of the IV - DV relationship -- so much of the IV could be counted on to produce so much of the DV.tons manure @ acre Bushel-yield @ acreRA & manipulate which plots receive how much manureIn behavioral research -- linear correlation &regression are most often applied to natural groups designs (often called correlational designs - but be careful to keep in mind that the important relationship is between the type of design and causal interpretability, not between the type of stat and causal interpretability)• A measured IV is used, so the value of the predictor is known with “more or less” precision (there is certainly some error)• This is usually not a true experiment, so causal interpretations are not made of the IV - DV relationship --so much IV leads to a prediction of so much DV.• Linear prediction is still dependent upon a linear relationship (if two things aren’t linearly related, a linear model can’t be used to predict one from the other)Two big differences between the non- & experimental uses of regression:1) precision of IV values 2) design & causal interpEach of these will show up as part of “statistical control”Reviewing Variations of statistical control via residualizationpartial correlation -- correlation between two variables (x & y) controlling both for some 3rdvariable (z)--ryx.zsemi-partial correlation -- correlation between two variables (part correlation) (x & y) controlling one of the variables for some 3rdvariable (z)--ry(x.z) & rx(y.z)Remember – the “3rdvariable” can be about causality,collinearity or measurement error Control of multiple variables…multiple partial correlation -- like partial, but with “multiple 3rdvariables”--ryx.zabcmultiple semi-partial correlation -- like semi-partial, but with “multiple 3rdvariables”--ry(x.zabc) vs. rx(y.zabc) ANCOVA -- ANalysis of COVAriance -- a kind of semi-partial ormultiple semi-partial corr-- test of the DV mean difference between IV conditions that is independent of some 3rdvariable (called the “covariate”)-- r DV(IV.cov)Statistical control is about the underlying causal model of the variables …XYrY,XZ3rdvariable problem …• relationship between x & y exists because a 3rdvariable is influencing both• Z might be a causal or a psychometric influenceXYZrY,XβxβzCombining the two…Including X changes apparent Z-Y relationship because Z influences both X & YXYrY,XZβzβxCollinearity problem …• by ignoring Z, the bivariate relationship mis-estimates X-Y relationship Statistical control is about correctiong for under-specification…When we take a bivariate look at part of a multivariate picture …• we’ll underestimate how much we can know about the criterion• we’ll likely mis-estimate how that predictor relates to the criterion• leaving predictors out usually leads to over-estimation r > β• leaving predictors out changes the collinearity structure, and so, might cause us to miss suppressor effects r < βStatistical control in an attempt to improve this …• what “would be” the bivariate relationship between these variables, in a population for which the control variable(s) is a constant (and so is not collinear with these variables) ?• it is very much like looking at the β for that predictor in a multiple regression, but is in the form of a “corrected” simple correlationr y(x.z) ≈βxfrom y’ = βxX+ βzZ ≈ R2y.X,Z–R2y.X“What is the relationship between y and the part of x that is independent of Z?Allman Brothers Band - No One To Run With.mp3Statistical control is about “correcting” the bivariate correlation to take the control variable(s) “into account”What do we get from this? Here’s where opinions differ …1. A substitute for experimantal control ?2. A better estimate of the causal relationship of the 2 variables ?3. A substitute for construct validity ?4. Solves under-specification problem ?5. Probably a better description of the relationship of these 2 variables than is the bivariate analysis ?Rejection of 1-4 (probably for good reasons) has led some to reject the 5thas well …but then what are we to do?• Only perform bivariate analyses (known to be flawed) ?• Expect to construct the “full story” from convergent research? (which is already the answer for “what’s the correct study”!!!)More complex models are, on average, more likely to be accurate!Apparent Bivariate Correlation- 0 + Bivariate Correlation after Correctionfor 3rdVariable- 0 +Some examples of taking the 3rdvariable into accountOf course, it can be uglier than that…Here, what “correction” you get depends upon which variable you control for –remember, larger models are only more accurate “on average”Here, the addition of the “Z” variable