1-Way Random Effects Model

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Presentation transcript:

1-Way Random Effects Model NBA Player Efficiencies by Game 2016-2017 Regular Season

Setting Population of Units Each Unit has a True Mean Level of Response Individual Measurements Within Units Vary Around the Unit’s Mean Response Overall Population Mean Response is Mean of Individual Units’ Mean Responses A Sample of g Units are Taken, and n Measurements are Made Among a Sample of ni “Subunits” Within the ith Sampled Unit. Goals: Estimate Population Mean Response, Among and Within Unit Variances, Intraclass Correlation

Statistical Model – Normal Effects and Errors - I

Means and Variances of Sample Means

Sums of Squares and Their Expectations

ANOVA F-test – Balanced Case

Estimating Overall Mean m - Balanced Data

Estimating Intra-Class Correlation rI = sa2 /( sa2 + s2)

Estimating Within Group Variance: s2

Estimating Between Group Variance: sa2

Example – NBA Player Efficiencies – 2016/7 Population: 409 Players (At least 10 games with more than 6 minutes of playing time in a game) Total of 23655 player games (Average of 57.8 games per player) Response Variable: Efficiency scaled to 36 Minutes: 36[(Points+Rebounds+Assists+Steals+Blocks)-(MissedFG+MissedFT+Turnovers)]/Minutes Obtain Average for each player Obtain (Unweighted) Average across players Obtain player effect (Player Average – Overall Average) Obtain within player “Errors” (Game – Player Average)

Population Results

Sampling/Analysis Procedure Take many random samples (10000) Within Samples: Take a Random Sample of Players (g=15) For each Sampled Player, Take a Random Sample of Games (n=6) Compute and Save: F-Statistic Estimates and Confidence Intervals for: Mean, Variance Components, and Intraclass Correlation Compare with True Values of Parameters

Numeric Results Notes: Between variance estimates are biased low (below the true value) partly caused by negative estimates Coverage rates are below the nominal .95 due to the fact that the random player effects are skewed

Histogram of the Estimates of m

Histogram of the (Scaled) Estimates of s2

Histogram of the Estimates of sa2

Histogram of Estimates of rI

Histogram of F