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Analyzing Data from Small N Designs using Multilevel Models Eden Nagler The Graduate Center, CUNY David Rindskopf, Ph.D The Graduate Center, CUNY

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2 Overview/Intro What is our current work? Where did we start? How does HLM fit into this framework?

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3 2 Initial Datasets: Stuart, R.B. (1967). Behavioral control of overeating. Behavior Research & Therapy, 5, ( ). 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), ( ).

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4 Stuart (1967):

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5 Stuart (1967): Procedures for Getting data into HLM

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7 Stuart (1967): Level-1 dataset

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8 Stuart (1967): Level-2 dataset

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9 Stuart (1967): HLM (Linear model) Linear Model: POUNDS = π 0 + π 1 *(MONTHS12) + e

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10 Stuart (1967): HLM – Linear Model Estimates Final estimation of fixed effects: Standard Approx. Fixed Effect CoefficientError T-ratiod.f.P-value For INTRCPT1,P0 INTRCPT2, B For MONTHS12 slope, P1 INTRCPT2, B The outcome variable is POUNDS POUNDS ij ≈ – 3.1*(MONTHS12) + e ij

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11 Stuart (1967): HLM – Quadratic Model Quadratic Model: POUNDS = π 0 + π 1 *(MONTHS12)+ π 2 *(MON12SQ)+e

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12 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, B For MONTHS12 slope, P1 INTRCPT2, B For MON12SQ slope, P2 INTRCPT2, B The outcome variable is POUNDS POUNDS ij ≈ – 1.8(MONTHS12) + 0.1*(MON12SQ) + e ij

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13 Stuart (1967): HLM – Linear vs. Quadratic Model Stuart (1967) – Actual Data Quadratic Model Prediction Linear Model Prediction

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14 Dicarlo & Reid (2004):

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15 Dicarlo & Reid (2004): Level-1 dataset

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16 Dicarlo & Reid (2004): Level-2 dataset

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17 Dicarlo & Reid (2004): HLM – Simple Model Simple Model: FREQRND = π 0 + π 1 *(PHASE) + e

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18 Dicarlo & Reid (2004): HLM – Simple Model Estimates Level-1 ModelLevel-2 Model log[L] = P0 + P1*(PHASE) P0 = B00 + R0 P1 = B10 + R Final estimation of fixed effects: (Unit-specific model) Standard Approx. Fixed Effect Coefficient Error T-ratiod.f.P-value For INTRCPT1,P0 INTRCPT2, B For PHASE slope,P1 INTRCPT2, B LN(FREQRND ij ) = *(PHASE) + e ij

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19 Dicarlo & Reid (2004): HLM – Simple Model Estimates LOG(FREQRND ij ) = B 00 + B 10 *(PHASE) + e ij For PHASE=0 (BASELINE): LOG(FREQRND ij ) = B 00 FREQRND ij = exp(B 00 ) For PHASE=1 (TREATMENT): LOG(FREQRND ij ) = B 00 + B 10 FREQRND ij = exp(B 00 +B 10 ) = exp(B 00 )*exp(B 10 ) Estimates: B 00 = -0.77; B 10 = 2.52 For PHASE=0 (BASELINE): FREQRND ij = exp(B 00 ) = exp(-0.77) = 0.46 For PHASE=1 (TREATMENT): FREQRND ij = exp(B 00 +B 10 ) = exp( ) = exp(1.75) = 5.75

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20 In conclusion… 1. Other issues we’ve encountered and explored 2. Issues we’ve encountered, but not yet explored 3. Issues we’ve not yet encountered nor explored

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