I have a linear model where my response Y is say the percentage (proportion) of fat in milk. I have two explanatory variables one (x1) is a continuous variable, the other (z) is a three level factor.
I now do the regression in R as:
contrasts(z) <- "contr.sum"
model<-lm(logit(Y) ~ log(x1)*z)
the model summary gives me the R2 of this model . However, I want to find out the importance of x1 in my model.
I can look at the p-value if the slope is statistically different from 0, but this does not tell me if x1 is actually a good predictor.
Is there a way to get the partial R2 for this model and the overall effect of x1? As this model includes an interaction I am not sure how to calculate this and if there is one unique solution or if I get a partial R2 for the main effect of x1 and a partial R2 for main effect of x1 plus its interaction.
Or would it be better to avoid partial R2 and explain the magnitude of the slope of the main effect and interaction. But given my logit transformation I am not sure if this has any practical meaning for say how log(x1) changes the log odds ratio of % fat in milk.
Thanks.
-I tried to fit the model without the interaction and without the factor to get a usual R2 , but this would not be my preferred solution and I would like to get the partial R2 when specifying a full model.
Update: As requested in a comment, here the output from the summary(model). As written above z is sum contrast coded.
Call:
lm(formula = y ~ log(x1) * z, data = mydata)
Residuals:
Min 1Q Median 3Q Max
-1.21240 -0.09487 0.03282 0.13588 0.85941
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.330678 0.034043 -68.462 < 2e-16 ***
log(x1) -0.012948 0.005744 -2.254 0.02454 *
z1 0.140710 0.048096 2.926 0.00357 **
z2 -0.348526 0.055156 -6.319 5.17e-10 ***
log(x1):z1 0.017051 0.008095 2.106 0.03558 *
log(x1):z2 -0.028201 0.009563 -2.949 0.00331 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2288 on 594 degrees of freedom
Multiple R-squared: 0.1388, Adjusted R-squared: 0.1315
F-statistic: 19.15 on 5 and 594 DF, p-value: < 2.2e-16
Update: As requested in a comment, here the output from
print(aov(model))
Call:
aov(formula = model)
Terms:
log(x1) z log(x1):z Residuals
Sum of Squares 0.725230 3.831223 0.456677 31.105088
Deg. of Freedom 1 2 2 594
Residual standard error: 0.228835
Estimated effects may be unbalanced.
As written above, z is sum contrast coded.
I used a linear model to obtain the best fit to my data, lm() function.
From literature I know that the optimal fit would be a linear regression with the slope = 1 and the intercept = 0. I would like to see how good this equation (y=x) fits my data? How do I proceed in order to find an R^2 as well as a p-value?
This is my data
(y = modelled, x = measured)
measured<-c(67.39369,28.73695,60.18499,49.32405,166.39318,222.29022,271.83573,241.72247, 368.46304,220.27018,169.92343,56.49579,38.18381,49.33753,130.91752,161.63536,294.14740,363.91029,358.32905,239.84112,129.65078,32.76462,30.13952,52.83656,67.35427,132.23034,366.87857,247.40125,273.19316,278.27902,123.24256,45.98363,83.50199,240.99459,266.95707,308.69814,228.34256,220.51319,83.97942,58.32171,57.93815,94.64370,264.78007,274.25863,245.72940,155.41777,77.45236,70.44223,104.22838,294.01645,312.42321,122.80831,41.65770,242.22661,300.07147,291.59902,230.54478,89.42498,55.81760,55.60525,111.64263,305.76432,264.27192,233.28214,192.75603,75.60803,63.75376)
modelled<-c(42.58318,71.64667,111.08853,67.06974,156.47303,240.41188,238.25893,196.42247,404.28974,138.73164,116.73998,55.21672,82.71556,64.27752,145.84891,133.67465,295.01014,335.25432,253.01847,166.69241,68.84971,26.03600,45.04720,75.56405,109.55975,202.57084,288.52887,140.58476,152.20510,153.99427,75.70720,92.56287,144.93923,335.90871,NA,264.25732,141.93407,122.80440,83.23812,42.18676,107.97732,123.96824,270.52620,388.93979,308.35117,100.79047,127.70644,91.23133,162.53323,NA ,276.46554,100.79440,81.10756,272.17680,387.28700,208.29715,152.91548,62.54459,31.98732,74.26625,115.50051,324.91248,210.14204,168.29598,157.30373,45.76027,76.07370)
Now I would like to see how good the equation y=x fits the data presented above (R^2 and p-value)?
I am very grateful if somebody can help me with this (basic) problem, as I found no answers to my question on stackoverflow?
Best regards Cyril
Let's be clear what you are asking here. You have an existing model, which is "the modelled values are the expected value of the measured values", or in other words, measured = modelled + e, where e are the normally distributed residuals.
You say that the "optimal fit" should be a straight line with intercept 0 and slope 1, which is another way of saying the same thing.
The thing is, this "optimal fit" is not the optimal fit for your actual data, as we can easily see by doing:
summary(lm(measured ~ modelled))
#>
#> Call:
#> lm(formula = measured ~ modelled)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -103.328 -39.130 -4.881 40.428 114.829
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 23.09461 13.11026 1.762 0.083 .
#> modelled 0.91143 0.07052 12.924 <2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 55.13 on 63 degrees of freedom
#> Multiple R-squared: 0.7261, Adjusted R-squared: 0.7218
#> F-statistic: 167 on 1 and 63 DF, p-value: < 2.2e-16
This shows us the line that would produce the optimal fit to your data in terms of reducing the sum of the squared residuals.
But I guess what you are asking is "How well do my data fit the model measured = modelled + e ?"
Trying to coerce lm into giving you a fixed intercept and slope probably isn't the best way to answer this question. Remember, the p value for the slope only tells you whether the actual slope is significantly different from 0. The above model already confirms that. If you want to know the r-squared of measured = modelled + e, you just need to know the proportion of the variance of measured that is explained by modelled. In other words:
1 - var(measured - modelled) / var(measured)
#> [1] 0.7192672
This is pretty close to the r squared from the lm call.
I think you have sufficient evidence to say that your data is consistent with the model measured = modelled, in that the slope in the lm model includes the value 1 within its 95% confidence interval, and the intercept contains the value 0 within its 95% confidence interval.
As mentioned in the comments, you can use the lm() function, but this actually estimates the slope and intercept for you, whereas what you want is something different.
If slope = 1 and the intercept = 0, essentially you have a fit and your modelled is already the predicted value. You need the r-square from this fit. R squared is defined as:
R2 = MSS/TSS = (TSS − RSS)/TSS
See this link for definition of RSS and TSS.
We can only work with observations that are complete (non NA). So we calculate each of them:
TSS = nonNA = !is.na(modelled) & !is.na(measured)
# residuals from your prediction
RSS = sum((modelled[nonNA] - measured[nonNA])^2,na.rm=T)
# total residuals from data
TSS = sum((measured[nonNA] - mean(measured[nonNA]))^2,na.rm=T)
1 - RSS/TSS
[1] 0.7116585
If measured and modelled are supposed to represent the actual and fitted values of an undisclosed model, as discussed in the comments below another answer, then if fm is the lm object for that undisclosed model then
summary(fm)
will show the R^2 and p value of that model.
The R squared value can actually be calculated using only measured and modelled but the formula is different if there is or is not an intercept in the undisclosed model. The signs are that there is no intercept since if there were an intercept sum(modelled - measured, an.rm = TRUE) should be 0 but in fact it is far from it.
In any case R^2 and the p value are shown in the output of the summary(fm) where fm is the undisclosed linear model so there is no point in restricting the discussion to measured and modelled if you have the lm object of the undisclosed model.
For example, if the undisclosed model is the following then using the builtin CO2 data frame:
fm <- lm(uptake ~ Type + conc, CO2)
summary(fm)
we have the this output where the last two lines show R squared and p value.
Call:
lm(formula = uptake ~ Type + conc, data = CO2)
Residuals:
Min 1Q Median 3Q Max
-18.2145 -4.2549 0.5479 5.3048 12.9968
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 25.830052 1.579918 16.349 < 2e-16 ***
TypeMississippi -12.659524 1.544261 -8.198 3.06e-12 ***
conc 0.017731 0.002625 6.755 2.00e-09 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.077 on 81 degrees of freedom
Multiple R-squared: 0.5821, Adjusted R-squared: 0.5718
F-statistic: 56.42 on 2 and 81 DF, p-value: 4.498e-16
I would like to simulate data for a logistic regression where I can specify its explained variance beforehand. Have a look at the code below. I simulate four independent variables and specify that each logit coefficient should be of size log(2)=0.69. This works nicely, the explained variance (I report Cox & Snell's r2) is 0.34.
However, I need to specify the regression coefficients in such a way that a pre-specified r2 will result from the regression. So if I would like to produce an r2 of let's say exactly 0.1. How do the coefficients need to be specified? I am kind of struggling with this..
# Create independent variables
sigma.1 <- matrix(c(1,0.25,0.25,0.25,
0.25,1,0.25,0.25,
0.25,0.25,1,0.25,
0.25,0.25,0.25,1),nrow=4,ncol=4)
mu.1 <- rep(0,4)
n.obs <- 500000
library(MASS)
sample1 <- as.data.frame(mvrnorm(n = n.obs, mu.1, sigma.1, empirical=FALSE))
# Create latent continuous response variable
sample1$ystar <- 0 + log(2)*sample1$V1 + log(2)*sample1$V2 + log(2)*sample1$V3 + log(2)*sample1$V4
# Construct binary response variable
sample1$prob <- exp(sample1$ystar) / (1 + exp(sample1$ystar))
sample1$y <- rbinom(n.obs,size=1,prob=sample1$prob)
# Logistic regression
logreg <- glm(y ~ V1 + V2 + V3 + V4, data=sample1, family=binomial)
summary(logreg)
The output is:
Call:
glm(formula = y ~ V1 + V2 + V3 + V4, family = binomial, data = sample1)
Deviance Residuals:
Min 1Q Median 3Q Max
-3.7536 -0.7795 -0.0755 0.7813 3.3382
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.002098 0.003544 -0.592 0.554
V1 0.691034 0.004089 169.014 <2e-16 ***
V2 0.694052 0.004088 169.776 <2e-16 ***
V3 0.693222 0.004079 169.940 <2e-16 ***
V4 0.699091 0.004081 171.310 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 693146 on 499999 degrees of freedom
Residual deviance: 482506 on 499995 degrees of freedom
AIC: 482516
Number of Fisher Scoring iterations: 5
And Cox and Snell's r2 gives:
library(pscl)
pR2(logreg)["r2ML"]
> pR2(logreg)["r2ML"]
r2ML
0.3436523
If you add a random error term to the ystar variable making ystat.r and then work with that, you can tweek the standard deviation until it meets you specifications.
sample1$ystar.r <- sample1$ystar+rnorm(n.obs, 0, 3.8) # tried a few values
sample1$prob <- exp(sample1$ystar.r) / (1 + exp(sample1$ystar.r))
sample1$y <- rbinom(n.obs,size=1,prob=sample1$prob)
logreg <- glm(y ~ V1 + V2 + V3 + V4, data=sample1, family=binomial)
summary(logreg) # the estimates "shrink"
pR2(logreg)["r2ML"]
#-------
r2ML
0.1014792
R-squared (and its variations) is a random variable, as it depends on your simulated data. If you simulate data with the exact same parameters multiple times, you'll most likely get different values for R-squared each time. Therefore, you cannot produce a simulation where the R-squared will be exactly 0.1 just by controlling the parameters.
On the other hand, since it's a random variable, you could potentially simulate your data from a conditional distribution (conditioning on a fixed value of R-squared), but you would need to find out what these distributions look like (math might get really ugly here, cross validated is more appropriate for this part).
My R-script produces glm() coeffs below.
What is Poisson's lambda, then? It should be ~3.0 since that's what I used to create the distribution.
Call:
glm(formula = h_counts ~ ., family = poisson(link = log), data = pois_ideal_data)
Deviance Residuals:
Min 1Q Median 3Q Max
-22.726 -12.726 -8.624 6.405 18.515
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 8.222532 0.015100 544.53 <2e-16 ***
h_mids -0.363560 0.004393 -82.75 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 11451.0 on 10 degrees of freedom
Residual deviance: 1975.5 on 9 degrees of freedom
AIC: 2059
Number of Fisher Scoring iterations: 5
random_pois = rpois(10000,3)
h=hist(random_pois, breaks = 10)
mean(random_pois) #verifying that the mean is close to 3.
h_mids = h$mids
h_counts = h$counts
pois_ideal_data <- data.frame(h_mids, h_counts)
pois_ideal_model <- glm(h_counts ~ ., pois_ideal_data, family=poisson(link=log))
summary_ideal=summary(pois_ideal_model)
summary_ideal
What are you doing here???!!! You used a glm to fit a distribution???
Well, it is not impossible to do so, but it is done via this:
set.seed(0)
x <- rpois(10000,3)
fit <- glm(x ~ 1, family = poisson())
i.e., we fit data with an intercept-only regression model.
fit$fitted[1]
# 3.005
This is the same as:
mean(x)
# 3.005
It looks like you're trying to do a Poisson fit to aggregated or binned data; that's not what glm does. I took a quick look for canned ways of fitting distributions to canned data but couldn't find one; it looks like earlier versions of the bda package might have offered this, but not now.
At root, what you need to do is set up a negative log-likelihood function that computes (# counts)*prob(count|lambda) and minimize it using optim(); the solution given below using the bbmle package is a little more complex up-front but gives you added benefits like easily computing confidence intervals etc..
Set up data:
set.seed(101)
random_pois <- rpois(10000,3)
tt <- table(random_pois)
dd <- data.frame(counts=unname(c(tt)),
val=as.numeric(names(tt)))
Here I'm using table rather than hist because histograms on discrete data are fussy (having integer cutpoints often makes things confusing because you have to be careful about right- vs left-closure)
Set up density function for binned Poisson data (to work with bbmle's formula interface, the first argument must be called x, and it must have a log argument).
dpoisbin <- function(x,val,lambda,log=FALSE) {
probs <- dpois(val,lambda,log=TRUE)
r <- sum(x*probs)
if (log) r else exp(r)
}
Fit lambda (log link helps prevent numerical problems/warnings from negative lambda values):
library(bbmle)
m1 <- mle2(counts~dpoisbin(val,exp(loglambda)),
data=dd,
start=list(loglambda=0))
all.equal(unname(exp(coef(m1))),mean(random_pois),tol=1e-6) ## TRUE
exp(confint(m1))
## 2.5 % 97.5 %
## 2.972047 3.040009
How can you get R's glm() to match polynomial data? I've tried several iterations of 'family=AAA(link="BBB")' but I can't seem to get trivial predictions to match.
For example, please help with R's glm to match polynomial data
x=seq(-6,6,2)
y=x*x
parabola=data.frame(x,y)
plot(parabola)
model=glm(y~x,dat=parabola)
test=data.frame(x=seq(-5,5,2))
test$y=predict(model,test)
plot(test)
The plot(parabola) looks as expected, but I can find the incantation of glm() that will make plot(test) look parabolic.
I think you need to step back and start to think about a model and how you represent this in R. In your example, y is a quadratic function of x, so you need to include x and x^2 in the model formula, i.e. as predictors you need to estimate the effect of x and x^2 on the response given the data to hand.
If y is Gaussian, conditional upon the model, then you can do this with lm() and either
y ~ x + I(x^2)
or
y ~ poly(x, 2)
In the first, we wrap the quadratic term in I() as the ^ operator has a special meaning (not its mathematical one) in an R model formula. The latter version gives orthogonal polynomials and hence the x and x^2 terms won't be correlated which can help with fitting, however in some cases interpreting the coefficients is trickier with poly().
Putting it all together we have (note that I add some random error to y so as to not predict it perfectly as the example I use is more common in reality):
x <- seq(-6 ,6 ,2)
y <- x^2 + rnorm(length(x), sd = 2)
parabola <- data.frame(x = x, y = y)
mod <- lm(y ~ poly(x, 2), data = parabola)
plot(parabola)
lines(fitted(mod) ~ x, data = parabola, col = "red")
The plot produced is:
An additional issue is whether y is Gaussian? If y can't be negative (i.e. a count), and/or is discrete, modelling using lm() is going to be wrong. That's where glm() might come in, by which you might fit a curve without needing x^2 (although if the data really are a parabola, then x on its own isn't going to fit the response), as there is an explicit transformation of the data from the linear predictor on to the scale of the response.
It is better to think about the properties of the data and the sort of model you want to fit and then build up the degree of polynomial within that modelling framework, rather than jumping in a trying various incantations to simply curve fit the data.
The match is now perfect. A slightly more interesting parabola:
x=seq(-16,16,2)
y= 4*x*x + 10*x + 6
parabola=data.frame(x,y)
plot(parabola)
model=lm(y~poly(x,2),dat=parabola)
summary(model)
test=data.frame(x=seq(-15,15,2))
test$y=predict(model,test)
points(test,pch=3)
An amateur (like me) might expect the coefficients of the model to be (4,10,6) to match 4*x*x + 10*x + 6
Call:
lm(formula = y ~ poly(x, 2), data = parabola)
Residuals:
Min 1Q Median 3Q Max
-3.646e-13 -8.748e-14 -3.691e-14 4.929e-14 6.387e-13
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.900e+02 5.192e-14 7.511e+15 <2e-16 ***
poly(x, 2)1 4.040e+02 2.141e-13 1.887e+15 <2e-16 ***
poly(x, 2)2 1.409e+03 2.141e-13 6.581e+15 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.141e-13 on 14 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 2.343e+31 on 2 and 14 DF, p-value: < 2.2e-16
Why would the coefficients be (390,404,1409)?