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Methodology
Journal of Chinese Integrative Medicine: Volume 10, 2012   Issue 4
Estimation of sample size and testing power (Part 7)
1. Liang-ping Hu (Consulting Center of Biomedical Statistics, Academy of Military Medical Sciences, Beijing 100850, China E-mail: E-mail: lphu812@sina.com)
2. Xiao-lei Bao (Consulting Center of Biomedical Statistics, Academy of Military Medical Sciences, Beijing 100850, China )
3. Xue Guan (Consulting Center of Biomedical Statistics, Academy of Military Medical Sciences, Beijing 100850, China )
ABSTRACT: Two-factor factorial design refers to the research involving two experimental factors and the number of the experimental groups equals to the product of the levels of the two experimental factors. In other words, it is the complete combination of the levels of the two experimental factors. The research subjects are randomly divided into the experimental groups. The two experimental factors are performed on the subjects at the same time, meaning that there is no order. The two experimental factors are equal during statistical analysis, that is to say, there is no primary or secondary distinction, nor nested relation. This article introduces estimation of sample size and testing power of quantitative data with two-factor factorial design.

Received February 10, 2012; accepted February 22, 2012; published online April 15, 2011.
Full-text LinkOut at PubMed. Journal title in PubMed: Zhong Xi Yi Jie He Xue Bao.

基金项目:国家科技重大专项“重大新药创制”资助项目 (No. 2009ZX09502-028)
Correspondence: Prof. Liang-ping Hu; Tel: 010-66931130; E-mail: lphu812@sina.com

  


     Two-factor factorial design is frequently adopted when the research involves two experimental factors, which are performed on the research subjects at the same time and are equal, and have level 1 interaction which is required to be examined. When there is only one quantitative index, it is called univariate quantitative data with two-factor factorial design, which can be analyzed by the analysis of variance of univariate quantitative data with two-factor factorial design[1]. This article introduces the estimation of sample size and testing power of univariate quantitative data with two-factor factorial design. Since the estimation of sample size and testing power of two-factor factorial design is complicated, this article will introduce it through examples when performing the analysis of variance by SAS.

 
  


1  Estimation of sample size
1.1  Example 1 
A researcher planed to conduct an animal experiment to study the influence of copper (Cu) and vitamin E (VE) on liver injury caused by carbon tetrachloride (CCl4). Thirty healthy male Wistar rats were chosen for pilot test and were equally divided into 10 groups. How many rats were needed to perform the analysis of variance of two-factor factorial design in the formal experiment if the testing power was required to be 80%[2]?
1.2  Analysis  Step 1: Based on the information of the pilot test, estimate the population means and the population standard deviations of superoxide dismutase (SOD) under all the experimental conditions. The corresponding SAS program named exam1_1.sas is as follows:
 

data step1;
do Cu=0.00,0.05,0.10,0.20,0.40;
do VE=0,150;
do i=1 to 3;
input SOD@@; output;
end;end;end;
cards;
343.8331.6318.1465.2457.7449.5
430.0417.0419.3584.5570.0576.7
448.1413.1454.4604.3531.4605.0
443.7412.3468.7485.6516.4485.9
474.6455.5464.4509.8552.0498.6
;
run;
ods html;
proc means mean std;
var SOD;
class Cu VE;
run;
ods html close;


     Program explanation: First, create a new data set named step1; then invoke the MEANS procedure to estimate the means and the standard deviations.
     Below is the main output:

Response Variable: SOD

Cu

VE

Sample size

Mean

Standard deviation

0

0

3

331.166 666 7

12.855 478 7

150

3

457.466 666 7

7.852 600 4

0.05

0

3

422.100 000 0

6.937 578 8

150

3

577.066 666 7

7.256 950 7

0.1

0

3

438.533 333 3

22.250 018 7

150

3

580.233 333 3

42.292 355 5

0.2

0

3

441.566 666 7

28.260 455 3

150

3

495.966 666 7

17.696 421 5

0.4

0

3

464.833 333 3

9.557 370 6

150

3

520.133 333 3

28.159 782 2


     Step 2: Estimate the sample size based on the result of step 1. The needed SAS program named exam1_2.sas is as follows:
 

data step2;
do Cu=0.00,0.05,0.10,0.20,0.40;
do VE=0,150;
input mean @@; output;
end;end;
cards;
331.1666667 457.4666667
422.1000000 577.0666667
438.5333333 580.2333333
441.5666667 495.9666667
464.8333333 520.1333333
;
run;
ods html;
proc glmpower data=step2;
class Cu VE;
model mean=Cu|VE;
power
stddev=6 to 42by 12
ntotal=.
power=0.80;
run;quit;
ods html close;


     Program explanation: First, specify the population means under all the experimental conditions. The “glmpower” procedure performs prospective power analysis. The statement “model mean=Cu|VE” can also be written as “model mean= Cu VE Cu*VE”. The option “stddev=6 to 42 by 12” specifies 4 population standard deviations, namely, 6, 18, 30 and 42 since step 1 has already computed the minimum standard deviation as 6.9, and the maximum as 42.3. The option “power=0.80” designates 0.8 as the testing power. The above parameter values can be altered under specific situations.
     The main output is as follows:

Fixed Scenario Elements

Dependent variable

Mean

Nominal Power

0.8

Alpha

0.05



Computed N Total

Index

Source

Std Dev

Test DF

Error DF

Actual Power

N Total

1

Cu

6

4

10

>0.999

20

2

Cu

18

4

10

>0.999

20

3

Cu

30

4

10

0.984

20

4

Cu

42

4

10

0.812

20

5

VE

6

1

10

>0.999

20

6

VE

18

1

10

>0.999

20

7

VE

30

1

10

>0.999

20

8

VE

42

1

10

>0.999

20

9

Cu*VE

6

4

10

>0.999

20

10

Cu*VE

18

4

10

0.940

20

11

Cu*VE

30

4

20

0.812

30

12

Cu*VE

42

4

50

0.877

60


     The above result presents the needed sample sizes with different standard deviations. When the population standard deviation is 42, the testing power to infer that Cu has a protective effect on liver injury caused by CCl4 is 81.2%, and the testing power to infer that VE has a protective effect on liver injury caused by CCl4 is 99.9% as long as the sample size reaches 20. When the sample size is 60, the testing power to infer that there is interaction between Cu and VE is 87.7%. Therefore, when the population standard deviation is 42, 60 rats are needed; that is to say, each group needs 6 rats at least since there are 10 groups.

  


2  Estimation of testing power
2.1  Example 2 
In example 1, suppose that the researcher increased the sample size to 60 and conducted the additional experiments. The researcher wondered whether the testing power was sufficient if adopting the analysis of variance of univariate quantitative data with two-factor factorial design[2].
2.2  Analysis  Step 1, based on the information of the pilot test, estimate the population means and the population standard deviations of SOD under all the experimental conditions. See program exam1_1.sas for reference. The main output is as follows:

Response Variable: SOD

Cu

VE

Sample size

Mean

Standard deviation

0

0

6

341.450 000 0

15.695 445 2

150

6

463.766 666 7

9.612 006 4

0.05

0

6

413.183 333 3

11.141 708 4

150

6

573.183 333 3

19.615 750 5

0.1

0

6

423.866 666 7

28.221 386 7

150

6

595.783 333 3

45.698 639 7

0.2

0

6

453.133 333 3

61.289 237 8

150

6

495.883 333 3

16.018 915 9

0.4

0

6

459.716 666 7

25.860 581 3

150

6

521.166 666 7

36.075 291 6



     Step 2: Estimate the testing power based on the above results. The corresponding SAS program named exam1_2.sas is as follows:
 

data step3;
do Cu=0.00,0.05,0.10,0.20,0.40;
do VE=0,150;
input mean @@; output;
end;end;
cards;
341.4500000 463.7666667
413.1833333 573.1833333
423.8666667 595.7833333
453.1333333 495.8833333
459.7166667 521.1666667
;
run;
ods html;
proc glmpower data=step3;
class Cu VE;
model mean=Cu|VE;
power
stddev=9.6 45.7
ntotal=60
power=.;
run;quit;
ods html close;


     Program explanation: First, specify the population means under all the experimental conditions. The glmpower procedure performs prospective power analysis. The option “stddev=9.645.7” designates 9.6 and 45.7 as the minimum and the maximum population standard deviations. The option “ntotal=60” designates 60 as the total sample size.
     Below is the main output:

Fixed Scenario Elements

Dependent variable

Mean

Total sample size

60

Alpha

0.05

Error degrees of freedom

50


Computed Power

Index

Source

Std Dev

Test DF

Power

1

Cu

9.6

4

>0.999

2

Cu

45.7

4

>0.999

3

VE

9.6

1

>0.999

4

VE

45.7

1

>0.999

5

Cu*VE

9.6

4

>0.999

6

Cu*VE

45.7

4

0.936


     The above result shows that when the sample size is 60, the testing power to infer that Cu has a protective effect on liver injury caused by CCl4 reaches 99.9% and 99.9%(S=9.6 and S=45.7), and the testing power to infer that VE has a protective effect on liver injury caused by CCl4 reaches 99.9% and 99.9%(S=9.6 and S=45.7). When the population standard deviation is 9.6, the testing power to infer that there is interaction between Cu and VE reaches 99.9%. When the population standard deviation is 45.7, the testing power reaches 93.65%.

  
References
1. Hu LP. The application of statistical triple-type theory in experimental designs[M]. Beijing: People’s Military Medical Press, 2006. 94. Chinese.
2. Hu LP. SAS research design and statistical analysis[M]. Beijing: People’s Medical Publishing House, 2010. 202-204, 223-225. Chinese.
  
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