# Presentation: 681955

## 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:

- 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):

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

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

## Stuart (1967): Level-1 dataset 

## Stuart (1967): Level-2 dataset

## Stuart (1967): HLM (Linear model)

- Linear Model:
- POUNDS = π0 + π1*(MONTHS12) + e

## 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

- Quadratic Model:
- POUNDS = π0+ π1*(MONTHS12)+ π2*(MON12SQ)+e

## 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) – Actual Data**

**Quadratic Model Prediction**

**Linear Model Prediction**

## Dicarlo & Reid (2004):		

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

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

## Dicarlo & Reid (2004): HLM – Simple Model

- Simple Model:
-  	FREQRND = π 0 + π1*(PHASE) + e

## 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 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...

- Other issues we’ve encountered and explored
- Issues we’ve encountered, but not yet explored
- Issues we’ve not yet encountered nor explored