Analyzing Data from Small N Designs using Multilevel Models Eden Nagler The Graduate Center, CUNY...
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Transcript of Analyzing Data from Small N Designs using Multilevel Models Eden Nagler The Graduate Center, CUNY...
Analyzing Data from Small N Designs using Multilevel Models
Eden NaglerThe Graduate Center, CUNY
David Rindskopf, Ph.DThe Graduate Center, CUNY
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Overview/Intro
What is our current work?
Where did we start?
How does HLM fit into this framework?
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2 Initial Datasets:
Stuart, R.B. (1967). Behavioral control of overeating. Behavior Research & Therapy, 5, (357-365).
Dicarlo, C.F. & Reid, D.H. (2004). Increasing pretend toy play of toddlers with disabilities in an inclusive setting. Journal of Applied Behavior Analysis, 37(2), (197-207).
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Stuart (1967): HLM – Linear Model Estimates
Final estimation of fixed effects:Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------For INTRCPT1,P0 INTRCPT2, B00 156.439560 5.053645 30.956 7 0.000For MONTHS12 slope, P1 INTRCPT2, B10 -3.078984 0.233772 13.171 7 0.000----------------------------------------------------------The outcome variable is POUNDS----------------------------------------------------------
POUNDSij ≈ 156.4 – 3.1*(MONTHS12) + eij
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Stuart (1967): HLM – Quadratic Model Estimates
Final estimation of fixed effects:Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value-----------------------------------------------------------For INTRCPT1, P0 INTRCPT2, B00 158.833791 5.321806 29.846 7 0.000For MONTHS12 slope, P1 INTRCPT2, B10 -1.773039 0.358651 -4.944 7 0.001For MON12SQ slope, P2 INTRCPT2, B20 0.108829 0.021467 5.070 7 0.001-----------------------------------------------------------The outcome variable is POUNDS-----------------------------------------------------------
POUNDSij ≈ 158.8 – 1.8(MONTHS12) + 0.1*(MON12SQ) + eij
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Stuart (1967): HLM – Linear vs. Quadratic Model
134.8
157.7
180.5
203.3
226.2
POUNDS
-12.60 -9.30 -6.00 -2.70 0.60
MONTHS12
-12.00 -9.00 -6.00 -3.00 0137.8
157.2
176.7
196.2
215.7
MONTHS12
POUNDS
-12.00 -9.00 -6.00 -3.00 0138.7
158.8
179.0
199.1
219.2
MONTHS12
POUNDS
Stuart (1967) – Actual Data
Quadratic Model Prediction
Linear Model Prediction
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Dicarlo & Reid (2004): HLM – Simple Model Estimates
Level-1 Model Level-2 Modellog[L] = P0 + P1*(PHASE) P0 = B00 + R0
P1 = B10 + R1----------------------------------------------------------Final estimation of fixed effects: (Unit-specific model)
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------- For INTRCPT1,P0 INTRCPT2, B00 -0.769384 0.634548 -1.212 4 0.292 For PHASE slope,P1 INTRCPT2, B10 2.516446 0.278095 9.049 4 0.000 ----------------------------------------------------------
LN(FREQRNDij) = -0.77 + 2.52*(PHASE) + eij
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Dicarlo & Reid (2004): HLM – Simple Model Estimates
LOG(FREQRNDij) = B00 + B10*(PHASE) + eij
For PHASE=0 (BASELINE):LOG(FREQRNDij) = B00
FREQRNDij= exp(B00)
For PHASE=1 (TREATMENT):LOG(FREQRNDij) = B00 + B10
FREQRNDij= exp(B00+B10) = exp(B00)*exp(B10)
Estimates: B00 = -0.77; B10 = 2.52For PHASE=0 (BASELINE):FREQRNDij= exp(B00) = exp(-0.77) = 0.46
For PHASE=1 (TREATMENT):FREQRNDij= exp(B00+B10) = exp(-0.77+2.52) = exp(1.75)
= 5.75