How to run svy: regress in R or get R-squared in R for complex survey data

To get the multiple R2 of a svyglm model, you could also use the summ function in the jtools package.

Because "svyglm" objects are also "lm" objects by default -- see class(model1) in your example -- you can just run summary.lm(model1) to get the R^2.

For a Gaussian glm (where the population parameter is the OLS parameter) you can just divide the dispersion parameter by the population variance and subtract from 1

Using one of the examples from the svyglm help page:

> data(api)
> dstrat<-svydesign(id=~1,strata=~stype, weights=~pw, data=apistrat, fpc=~fpc)
> api.reg <- svyglm(api.stu~enroll, design=dstrat)
> summary(api.reg)

Call:
svyglm(formula = api.stu ~ enroll, design = dstrat)

Survey design:
dstrat<-svydesign(id=~1,strata=~stype, weights=~pw, data=apistrat, fpc=~fpc)

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 13.34383   11.46399   1.164    0.246    
enroll       0.81454    0.02459  33.120   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 7331.633)

Number of Fisher Scoring iterations: 2

> nullmodel<-svyglm(api.stu~1,design=dstrat)
> summary(nullmodel)

Call:
svyglm(formula = api.stu ~ 1, design = dstrat)

Survey design:
dstrat<-svydesign(id=~1,strata=~stype, weights=~pw, data=apistrat, fpc=~fpc)

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   498.23      16.06   31.02   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 137086.3)

Number of Fisher Scoring iterations: 2

> 1-7331.633/137086.3
[1] 0.9465181

You could also get the null-model variance using svyvar

> svyvar(~api.stu,design=dstrat)
        variance    SE
api.stu   137086 19197

And in this case we have the whole population, so we can run lm on the population and compare the survey estimate of rsquared with the population value

> summary(lm(api.stu ~ enroll,data=apipop))

Call:
lm(formula = api.stu ~ enroll, data = apipop)

Residuals:
     Min       1Q   Median       3Q      Max 
-1021.20   -13.76     6.13    29.56   498.98 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 13.613245   1.953709   6.968 3.55e-12 ***
enroll       0.813556   0.002522 322.581  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 92.16 on 6155 degrees of freedom
  (37 observations deleted due to missingness)
Multiple R-squared:  0.9442,    Adjusted R-squared:  0.9441 
F-statistic: 1.041e+05 on 1 and 6155 DF,  p-value: < 2.2e-16

As an added bonus answer: if you want the Nagelkerke or Cox-Snell r-squared for binary or count data, there's a function psrsq

You can get the amount of explained variation just by looking at the residual variance of your model and the residual variance of a null model (model with the explanatory covariates removed, including just an intercept). So the formula you need is:

R2_var = 1 - residual_variance / residual_variance_of_a_null_model