Randomized Incomplete Block Design



Randomized Incomplete Block Design

Stepping and Heart Rate (Data are Missing By Design)

Data Source: DASL (Data and Stories Library) Carnegie Mellon Institute



Description: An experiment was conducted by students at The Ohio State University in the fall of 1993 to explore the nature of the relationship between a person's heart rate and the frequency at which that person stepped up and down on steps of various heights. The response variable, heart rate, was measured in beats per minute. There were two different step heights: 5.75 inches (coded as 0), and 11.5 inches (coded as 1). There were three rates of stepping: 14 steps/min. (coded as 0), 21 steps/min. (coded as 1), and 28 steps/min. (coded as 2). This resulted in six possible height/frequency combinations. Each subject performed 5 of the 6 possible combinations, in random order. Each subject performed the activity for three minutes. Subjects were kept on pace by the beat of an electric metronome. One experimenter counted the subject's pulse for 20 seconds before and after each trial. The subject always rested between trials until her or his heart rate returned to close to the beginning rate. Another experimenter kept track of the time spent stepping. Each subject was always measured and timed by the same pair of experimenters to reduce variability in the experiment. Each pair of experimenters/subject was treated as a block.

Number of cases: 30. (Not included in Lecture Notes).

Study Design: Table Entries are Order of Combination

| | | Combination | | | |

| | |(first value indicates Height of step, second indcates | | | |

| | |rate of stepping ) | | | |

| | |00 |01 |02 |10 |11 |12 |

|Subject/Exp | | | | | | | |

|Block |1 |4 |2 |5 |3 |1 |-- |

| |2 |1 |2 |5 |4 |-- |3 |

| |3 |-- |2 |4 |3 |1 |5 |

| |4 |4 |-- |1 |3 |5 |2 |

| |5 |1 |4 |2 |-- |3 |5 |

| |6 |4 |3 |-- |2 |5 |1 |

Variable Descriptions:

1. Order: the overall performance order of the trial

2. Block: the subject and experimenters' block number

3. Height: 0 if step at the low (5.75") height, 1 if at the high (11.5") height

4. Frequency: the rate of stepping. 0 if slow (14 steps/min), 1 if medium (21 steps/min), 2 if high (28 steps/min)

5. Rest_HR: the resting heart rate of the subject before a trial, in beats per minute

6. HR: the final heart rate of the subject after a trial, in beats per minute

Command File: Stepping.sas

data step;

input

Order Block Height Freq Rest_HR HR;

cards;

16 2 0 0 60 75

18 2 0 1 63 84

19 2 1 2 69 135

21 2 1 0 69 108

25 2 0 2 69 93

17 4 1 1 96 141

20 4 1 0 87 120

22 4 0 0 90 99

23 4 1 2 93 153

27 4 0 2 87 129

24 3 1 1 72 99

26 3 0 1 69 93

28 3 1 0 78 93

29 3 0 2 72 99

30 3 1 2 78 129

1 5 0 0 87 93

2 1 1 1 87 111

3 6 1 2 81 120

4 5 0 2 75 123

5 1 0 1 81 96

6 6 1 0 84 99

7 1 1 0 84 99

8 5 1 1 90 129

9 6 0 1 75 90

10 1 0 0 78 87

11 6 0 0 84 84

12 5 0 1 90 108

13 1 0 2 78 96

14 6 1 1 84 90

15 5 1 2 90 147

;

title "Lab Example2: Randomized Incomplete Block";

proc mixed data=step;

class block height freq;

model hr = height freq height*freq;

random int / subject=block vcorr;

lsmeans height*freq/ slice=height slice=freq;

estimate "Mean ht 0 freq 0" int 1 height 1 0 freq 1 0 0

height*freq 1 0 0 0 0 0;

estimate "Freq 0: ht 0 vs 1" height 1 -1

height*freq 1 0 0 -1 0 0; run;

Lab Example2: Randomized Incomplete Block

The Mixed Procedure

Model Information

Data Set WORK.STEP

Dependent Variable HR

Covariance Structure Variance Components

Subject Effect Block

Estimation Method REML

Residual Variance Method Profile

Fixed Effects SE Method Model-Based

Degrees of Freedom Method Containment

Class Level Information

Class Levels Values

Block 6 1 2 3 4 5 6

Height 2 0 1

Freq 3 0 1 2

Dimensions

Covariance Parameters 2

Columns in X 12

Columns in Z Per Subject 1

Subjects 6

Max Obs Per Subject 5

Number of Observations

Number of Observations Read 30

Number of Observations Used 30

Number of Observations Not Used 0

Iteration Evaluations -2 Res Log Like Criterion

0 1 204.33612092

1 1 183.52381478 0.00000000

Convergence criteria met.

Estimated V Correlation Matrix for Block 1

Row Col1 Col2 Col3 Col4 Col5

1 1.0000 0.7665 0.7665 0.7665 0.7665

2 0.7665 1.0000 0.7665 0.7665 0.7665

3 0.7665 0.7665 1.0000 0.7665 0.7665

4 0.7665 0.7665 0.7665 1.0000 0.7665

5 0.7665 0.7665 0.7665 0.7665 1.0000

Covariance Parameter Estimates

Cov Parm Subject Estimate

Intercept Block 149.57

Residual 45.5763

Fit Statistics

-2 Res Log Likelihood 183.5

AIC (smaller is better) 187.5

AICC (smaller is better) 188.1

BIC (smaller is better) 187.1

Type 3 Tests of Fixed Effects

Num Den

Effect DF DF F Value Pr > F

Height 1 19 74.85 ................
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