Analysis of Discrete Data

Graphical Exploration of Statistical InteractionsNick JacksonUniversity of Southern CaliforniaDepartment of Psychology10/25/201311OverviewWhat is Interaction?2-Way InteractionsCategorical X CategoricalContinuous X CategoricalContinuous X Continuous3-Way InteractionsCategorical X Continuous X ContinuousContinuous X Continuous X ContinuousTime in a Three-Way Interaction4-Way and beyond

22What is an Interaction?Equivalent Statements:When the relationship between X and Y depends on the levels of a third variable Z.Z modifies the effect of X on Y.X and Y s relationship is different at differing levels of ZAlso Called Moderation or Effect Modification. Moderation is a stupid term.Moderation (n): The avoidance of excess or extremes. Moderate (v): To make or become less extreme or intenseThose are kinda the opposite of what we mean when we say moderation in a statistical sense.

33What is an Interaction?4XYZAs SEM diagrams:XYX*ZZWhat is an Interaction?5XYZ=0Z=1Z Modifies the effect of X on YXYEffect of X on Y if we ignore ZLeaving out interactions can lead us to false beliefs about how two variables X and Y are related to eachotherInteractions are interesting in that they point us in the direction of causal variables.5Types of Interaction6Quantitative Interaction OnlyQualitative InteractionZ=0YZ=1X=0X=1X=0X=1Quantitative Interaction: Difference between X(0) and X(1) is significantly different between Z(0) and Z(1), though these differences are not qualitatively different (visually these things look to be about the same). This occurs as a result of substantial power. X*Z, p30) and having high cholesterol results in high BP.

Two-Way Interactions19Continuous X Continuous Interaction: 3D Mesh Plots (Matlab, Sigma Plot, R)Same data as before, same interpretation. Use 4-Corners

Why we generally dont use observed datanot smoothObserved DataMarginal Estimates Datas19Two-Way Interactions20

Continuous X Continuous Interaction:Useful for Non-linear continuous interactions (Response Surface Model) The 4 corners method before cannot be applied here, as it doesnt account Start at the edges of one variable and interpret the shape in comparison to the other edge. 20Three-Way Interactions21Now things get complicated. Variables W*X*Z used to predict Y.The Interaction of X*Z is different at differing levels of WOr X*W is different at differing levels of ZOr Z*W is different at differing levels of XOr relationship of X and Y is different according to the levels of W and Z etc.Substantially easier when one of X, W, or Z are categorical

Three-Way InteractionsSubstantially easier when one of X, W, or Z are categorical.so we pick a small range of values to predict one of the variables overtreating it as semi-discrete (Quartiles?)Often Time is the third variableInterested in if the interaction of X*Z change over Time (W)

22

Three-Way Interactions23Categorical X Continuous X Continuous Interaction:Sleep Medication (Y/N) * BMI * Pulse: Stratify on categorical var Sleep MedsThe interaction of BMI and Pulse exists for those on Sleep Medications only.Three-Way Interactions24

Another way to look at this is how the difference in Apnea between those on Sleep Medications versus Not changes depending upon the relationships of pulse and BMIThree-Way Interactions25Continuous X Continuous X Continuous Interaction:Glucose Level* BMI * Pulse: Stratify on Glucose

Asks the question: How does the interaction of Pulse and BMI change across levels of glucoseThree-Way Interactions26

Continuous X Continuous X Continuous Interaction:Glucose Level* BMI * Pulse: Look at how the slopes of Glucose on Apnea change.Asks the question: How does the relationship of Glucose to Apnea change across levels of BMI and pulse.Three-Way Interactions27What if we have time as our third variable?Same techniques, but perhaps in the future we wont be limited to just static graphs.

Interaction of BMI and Pulse on Apnea Score across TimePresenting Data in MotionEven better, lets do some of this:http://www.ted.com/talks/hans_rosling_reveals_new_insights_on_poverty.html28Four-Way Interactions and BeyondUnderstanding anything much more complex than a 3-way interaction is difficult without a good way to break down variables into categoriesClassification Techniques/Machine Leaning/Exploratory Data MiningCan take high-dimensional data and find homogenous groups based upon relationships of continuous/categorical variables.29Four-Way Interactions and Beyond30Lateral Walls0.644

Soft Palate-1.845

Genioglossus-1.123

Mandibular Width-0.250

19.0 12.341.2 19.127.8 13.842.2 17.950.9 21.4Smaller StructureLargerStructureCART Model:4-Way Interaction of continuous variables on Apnea SeverityTake Home PointsTest for interactions in the beginning of model buildingCause they are interestingCause they obscure your main effectsInteractions give us clues about underlying etiology (David Schwartz). It is not enough to detect them, we have to understand why the interaction exists. We must search for the variable(s) that make interactions go away (mediated moderation)Modern classification/Data Mining Methods are great at detecting high-dimensional (numerous variables) non-linear interactions Stata Version 12 and 13 are amazing at doing these types of plots (margin plots). Also, check out Interpreting and Visualizing Regression Models Using Stata by Michael Mitchell

31