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

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analyzing data from small n designs using multilevel models

Analyzing Data from Small N Designs using Multilevel Models

Eden Nagler

The Graduate Center, CUNY

David Rindskopf, Ph.D

The Graduate Center, CUNY

overview intro
Overview/Intro
  • What is our current work?
  • Where did we start?
  • How does HLM fit into this framework?
2 initial datasets
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).

stuart 1967 hlm linear model
Stuart (1967): HLM (Linear model)

Linear Model:

POUNDS = π0 + π1*(MONTHS12) + e

stuart 1967 hlm linear model estimates
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 0.000

For 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

stuart 1967 hlm quadratic model
Stuart (1967): HLM – Quadratic Model

Quadratic Model:

POUNDS = π0+ π1*(MONTHS12)+ π2*(MON12SQ)+e

stuart 1967 hlm quadratic model estimates
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 0.000

For MONTHS12 slope, P1

INTRCPT2, B10 -1.773039 0.358651 -4.944 7 0.001

For 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

stuart 1967 hlm linear vs quadratic model
Stuart (1967): HLM – Linear vs. Quadratic Model

Stuart (1967) – Actual Data

Linear Model Prediction

Quadratic Model Prediction

dicarlo reid 2004 hlm simple model
Dicarlo & Reid (2004): HLM – Simple Model

Simple Model:

FREQRND = π 0 + π1*(PHASE) + e

dicarlo reid 2004 hlm simple model estimates
Dicarlo & Reid (2004): HLM – Simple Model Estimates

Level-1 Model Level-2 Model

log[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

dicarlo reid 2004 hlm simple model estimates1
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 = 2.52

For 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

in conclusion
In conclusion…
  • Other issues we’ve encountered and explored
  • Issues we’ve encountered, but not yet explored
  • Issues we’ve not yet encountered nor explored
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