Computing R squared with missing data

3 replies [Last post]
fife's picture
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Joined: 07/01/2010

This may be an easy question, but I can't think of the answer. I've built a regression model (predicting Y) with two independent variables (X and Z). I want to compute R^2. Is that built into the model somehow? If not, any ideas on how to compute it? Here's the model I have:

multiRegModel <- mxModel("Multiple Regression, All Variables",
type="RAM",
manifestVars=c("x", "y", "z"),
# variance paths
mxPath(
from=c("x", "y", "z"),
arrows=2,
free=TRUE,
values = c(.5, .5, .5),
labels=c("varx", "residual", "varz")
),
# covariance of x and z
mxPath(
from="x",
to="z",
arrows=2,
free=TRUE,
values=0.2,
labels="covxz"
),
# regression weights
mxPath(
from=c("x","z"),
to="y",
arrows=1,
free=TRUE,
values=.5,
labels=c("betax","betaz")
),
# means and intercepts
mxPath(
from="one",
to=c("x", "y", "z"),
arrows=1,
free=TRUE,
values=c(.5, .5),
labels=c("meanx", "beta0", "meanz")
)
) # close model

I know "residual" is the residual variance of Y, but I would think I'd need an estimate of the total variance of Y to compute it from that. Any help would be appreciated. Thanks!

fife's picture
Offline
Joined: 07/01/2010
not quite

Thanks for the input Ryne. I'm still having trouble. Below I have a reproducible example. I expected openmx and lm to agree on the value of R squared, but they don't. I've tried doing it with and without the covariance between x and z and they still don't agree. Any help would be appreciated.

#### simulate data
n = 10000
pop.mat = matrix(c(1, .48, .36, .48, 1, .35, .36, .35, 1), nrow=3)
d = data.frame(mvrnorm(n, mu=rep(0, times=3), Sigma=pop.mat))
names(d) = c(letters[c(26, 25, 24)])

### build openMX model
multiRegModel <- mxModel("MR",
type="RAM",
manifestVars=c("x", "y", "z"),
# variance paths
mxPath(
from=c("x", "y", "z"),
arrows=2,
free=TRUE,
values = c(.5, .5, .5),
labels=c("varx", "residual", "varz")
),
# covariance of x and z
mxPath(
from="x",
to="z",
arrows=2,
free=TRUE,
values=0.2,
labels="covxz"
),
# regression weights
mxPath(
from=c("x","z"),
to="y",
arrows=1,
free=TRUE,
values=.5,
labels=c("betax","betaz")
),
# means and intercepts
mxPath(
from="one",
to=c("x", "y", "z"),
arrows=1,
free=TRUE,
values=c(.5, .5),
labels=c("meanx", "beta0", "meanz")
)
) # close model

### estimate the model
m = mxModel(multiRegModel, mxData(observed=d, type="raw"))
mod = mxRun(model=m, silent=TRUE, suppressWarnings=TRUE)

### compute R squared
par = summary(mod)$parameters[,c("name","Estimate")]
res = par[par$name=="residual",2]
betax = par[par$name=="betax",2]
varx = par[par$name=="varx",2]
betaz = par[par$name=="betaz",2]
varz = par[par$name=="varz",2]
covxz = par[par$name=="covxz",2]

SST = (res + betax^2*varx + betaz^2*varz + betax*betaz*covxz)*(n-1)
SSR = res*(n-3)
Rsq = 1-(SSR/SST)

regMod = lm(y~x + z, data=d)

summary(regMod)$r.squared
Rsq

SST = (res + betax^2*varx + betaz^2*varz + betax*betaz*covxz)*(n-1)
SSR = res*(n-3)
Rsq.1 = 1-(SSR/SST)

par = summary(mod2)$parameters[,c("name","Estimate")]
res = par[par$name=="residual",2]
betaz = par[par$name=="betaz",2]
varz = par[par$name=="varz",2]

SST2 = (res + betaz^2*varz)*(n-1)
SSR2 = res*(n-2)
Rsq.2 = 1-(SSR2/SST2)

partial = sqrt((Rsq.1-Rsq.2)/(1-Rsq.2^2))

list(partial = partial, rsq1 = Rsq.1, rsq2=Rsq.2, SST=SST, SSR=SSR)

mhunter's picture
Offline
Joined: 07/31/2009
A different method

Adding a few lines to your example gives me numbers that match to my satisfaction (4-6 decimals).

resid.mod <- omxGetParameters(mod)["residual"]
var.y <- var(d$y) #total var of y
(modR2 <- (var.y - resid.mod)/var.y)
# 0.2579765
 
# total estimated var of y from model
var.ymod <- attr(mod@output$algebras$MR.fitfunction, "expCov")[2,2] 
(allmodR2 <- (var.ymod - resid.mod)/var.ymod)
# 0.2579022
 
summary(regMod)$r.squared
# 0.2579023

What do you think?

Ryne's picture
Offline
Joined: 07/31/2009
Your output has an expected

Your output has an expected covariance in the output slot (model@output$algebras$modelname.fitfunction, subbing model and modelname as necessary). 1-resid/total should yield R2

Alternatively, you could figure it out by hand if that's easier. Total variance of Y in this model would be:

Var(Y) = residual + betax^2 varx + betaz^2 varz + betax betaz covxz

There's probably an easy function to write, at least for RAM models.