在R和Stata当中计算回归系数效应量的方法

[chengjun]

HOW CAN I COMPUTE EFFECT SIZE IN STATA FOR REGRESSION?

https://stats.oarc.ucla.edu/stata/faq/how-can-i-compute-effect-size-in-stata-for-regression/

Two of the more common measures of effect size for regression analysis are eta2 and partial eta2.

• Eta2 is the proportion of the total variance that is attributed to an effect or set of effects.
• Partial eta2 is the proportion of effect and error variance that is attributable to the effect.

The formula differs from the eta squared formula in that the denominator includes the SSEffect plus the SSError rather than the SSTotal. The Stata regress postestimation command estat esize can be used to estimate eta2 for the model and partial eta2 for each effect in the model. Below, we run a linear regression analysis the hsbdemo dataset.

use https://stats.idre.ucla.edu/stat/data/hsbdemo, clear

(highschool and beyond (200 cases))

describe

Contains data from https://stats.idre.ucla.edu/stat/data/hsbdemo.dta
 Observations:           200                  highschool and beyond (200 cases)
    Variables:            13                  30 Oct 2009 14:13
--------------------------------------------------------------------------------
Variable      Storage   Display    Value
    name         type    format    label      Variable label
--------------------------------------------------------------------------------
id              float   %9.0g                 
female          float   %9.0g      fl         
ses             float   %9.0g      sl         
schtyp          float   %9.0g      scl        type of school
prog            float   %9.0g      sel        type of program
read            float   %9.0g                 reading score
write           float   %9.0g                 writing score
math            float   %9.0g                 math score
science         float   %9.0g                 science score
socst           float   %9.0g                 social studies score
honors          float   %19.0g     honlab     honors english
awards          float   %9.0g                 
cid             int     %8.0g                 
--------------------------------------------------------------------------------
Sorted by: 

regress write i.female read math i.prog

      Source |       SS           df       MS      Number of obs   =       200
-------------+----------------------------------   F(5, 194)       =     45.01
       Model |  9602.28627         5  1920.45725   Prob > F        =    0.0000
    Residual |  8276.58873       194  42.6628285   R-squared       =    0.5371
-------------+----------------------------------   Adj R-squared   =    0.5251
       Total |   17878.875       199   89.843593   Root MSE        =    6.5317

------------------------------------------------------------------------------
       write | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      female |
     female  |   5.384982    .929572     5.79   0.000     3.551617    7.218346
        read |   .3069424   .0611262     5.02   0.000     .1863852    .4274996
        math |   .3603705   .0690064     5.22   0.000     .2242715    .4964695
             |
        prog |
   academic  |    .436372   1.230379     0.35   0.723    -1.990265    2.863009
   vocation  |  -2.219748   1.359353    -1.63   0.104    -4.900756    .4612603
             |
       _cons |   15.16272   3.225088     4.70   0.000     8.801985    21.52346
------------------------------------------------------------------------------

We follow the regress command with estat esize, which displays estimates and confidence intervals for eta2 for the model and partial eta2 for each effect in the model.

estat esize

Effect sizes for linear models

-----------------------------------------------------------------
             Source | Eta-squared     df     [95% conf. interval]
--------------------+--------------------------------------------
              Model |   .5370744       5      .433662    .6003297
                    |
             female |   .1474719       1     .0667184    .2382202
               read |    .115024       1      .043701    .2017348
               math |   .1232518       1     .0493088    .2111699
               prog |   .0232415       2            .    .0732192
-----------------------------------------------------------------
Note: Eta-squared values for individual model terms are partial.

An anova table of this regression allows us to see how eta2 and partial eta2 are calculated.

anova write i.female c.read c.math i.prog

                         Number of obs =        200    R-squared     =  0.5371
                         Root MSE      =    6.53168    Adj R-squared =  0.5251

                  Source | Partial SS         df         MS        F    Prob>F
              -----------+----------------------------------------------------
                   Model |  9602.2863          5   1920.4573     45.01  0.0000
                         |
                  female |     1431.7          1      1431.7     33.56  0.0000
                    read |   1075.743          1    1075.743     25.21  0.0000
                    math |  1163.5091          1   1163.5091     27.27  0.0000
                    prog |  196.93763          2   98.468814      2.31  0.1022
                         |
                Residual |  8276.5887        194   42.662829  
              -----------+----------------------------------------------------
                   Total |  17878.875        199   89.843593  

The model eta2 is SSModel/SStotal = 9602.2863/17878.875 = 0.53707441. This matches the estimated for R-squared. The familiar interpretation is that the model explains 53.71% of the total variance of write.

Each partial eta2 is SSEffect/(SSEffect+SSError). The SSError for all of these terms is SSResidual. We can thus calculate partial eta2 for female = SSEffect/(SSEffect+SSError) = 1431.7/(1431.7+8276.5887) = 0.14747192.

We can interpret this to mean that about 14.75% of the variance unexplained by effects other than female is explained by the female effect. If we need estimates of eta2 for each effect, it is simply SSEffect/SSTotal.

[chengjun]

如何在R里计算effect size？

第一步，读入数据

同样使用上面stata使用的示例数据。使用foreign包当中的read.dta函数可以读入这个数据。

library(foreign)
library(effectsize)
library(car)

Loading required package: carData

dat = read.dta('https://stats.idre.ucla.edu/stat/data/hsbdemo.dta')

head(dat)

A data.frame: 6 × 13

	id	female	ses	schtyp	prog	read	write	math	science	socst	honors	awards	cid
	<dbl>	<fct>	<fct>	<fct>	<fct>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<fct>	<dbl>	<int>
1	45	female	low	public	vocation	34	35	41	29	26	not enrolled	0	1
2	108	male	middle	public	general	34	33	41	36	36	not enrolled	0	1
3	15	male	high	public	vocation	39	39	44	26	42	not enrolled	0	1
4	67	male	low	public	vocation	37	37	42	33	32	not enrolled	0	1
5	153	male	middle	public	vocation	39	31	40	39	51	not enrolled	0	1
6	51	female	high	public	general	42	36	42	31	39	not enrolled	0	1

第二步，跑回归分析

m = lm(write ~ female + read + math + prog, data = dat)
summary(m)

Call:
lm(formula = write ~ female + read + math + prog, data = dat)

Residuals:
     Min       1Q   Median       3Q      Max 
-17.5954  -3.6251   0.9111   3.9146  13.5497 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  15.16272    3.22509   4.701 4.89e-06 ***
femalefemale  5.38498    0.92957   5.793 2.75e-08 ***
read          0.30694    0.06113   5.021 1.16e-06 ***
math          0.36037    0.06901   5.222 4.53e-07 ***
progacademic  0.43637    1.23038   0.355    0.723    
progvocation -2.21975    1.35935  -1.633    0.104    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.532 on 194 degrees of freedom
Multiple R-squared:  0.5371,	Adjusted R-squared:  0.5251 
F-statistic: 45.01 on 5 and 194 DF,  p-value: < 2.2e-16

第三步，看标准化回归系数
标准化回归系数也是常见的effecsize表示方法

standardize_parameters(m)

A parameters_standardized: 6 × 5

Parameter	Std_Coefficient	CI	CI_low	CI_high
<chr>	<dbl>	<dbl>	<dbl>	<dbl>
(Intercept)	-0.27524921	0.95	-0.5040898	-0.04640862
femalefemale	0.56812078	0.95	0.3746990	0.76154253
read	0.33201800	0.95	0.2016119	0.46242413
math	0.35618306	0.95	0.2216655	0.49070064
progacademic	0.04603767	0.95	-0.2099749	0.30205020
progvocation	-0.23418556	0.95	-0.5170345	0.04866341

第四步，跑方差分析并计算partical eta squared

aov_table <- car::Anova(m, type = 3)
aov_table

A anova: 6 × 4

	Sum Sq	Df	F value	Pr(>F)
	<dbl>	<dbl>	<dbl>	<dbl>
(Intercept)	943.0196	1	22.10401	4.890944e-06
female	1431.7000	1	33.55849	2.753470e-08
read	1075.7430	1	25.21500	1.158227e-06
math	1163.5091	1	27.27220	4.533124e-07
prog	196.9376	2	2.30807	1.021780e-01
Residuals	8276.5887	194	NA	NA

eta_squared(aov_table)

A effectsize_anova: 4 × 5

Parameter	Eta2_partial	CI	CI_low	CI_high
<chr>	<dbl>	<dbl>	<dbl>	<dbl>
female	0.14747193	0.95	0.07871957	1
read	0.11502405	0.95	0.05369511	1
math	0.12325180	0.95	0.05984788	1
prog	0.02324152	0.95	0.00000000	1

[chengjun]

了解更多 在R和Stata当中计算回归系数效应量的方法 https://www.douban.com/group/topic/267379314