# Interpreting spline results

I'm trying to fit a spline for a GLM using R. Once I fit the spline, I want to be able to take my resulting model and create a modeling file in an Excel workbook.

For example, let's say I have a data set where y is a random function of x and the slope changes abruptly at a specific point (in this case @ x=500).

set.seed(1066)
x<- 1:1000
y<- rep(0,1000)

y[1:500]<- pmax(x[1:500]+(runif(500)-.5)*67*500/pmax(x[1:500],100),0.01)
y[501:1000]<-500+x[501:1000]^1.05*(runif(500)-.5)/7.5

df<-as.data.frame(cbind(x,y))

plot(df)


I now fit this using

library(splines)


and my results show

summary(spline1)

Call:
glm(formula = y ~ ns(x, knots = c(500)), family = Gamma(link = "log"),
data = df)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-4.0849  -0.1124  -0.0111   0.0988   1.1346

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)             4.17460    0.02994  139.43   <2e-16 ***
ns(x, knots = c(500))1  3.83042    0.06700   57.17   <2e-16 ***
ns(x, knots = c(500))2  0.71388    0.03644   19.59   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Gamma family taken to be 0.1108924)

Null deviance: 916.12  on 999  degrees of freedom
Residual deviance: 621.29  on 997  degrees of freedom
AIC: 13423

Number of Fisher Scoring iterations: 9


At this point, I can use predict function within r and get perfectly acceptable answers. The problem is that I want to use the model results to build a workbook in Excel.

My understanding of the predict function is that given a new "x" value, r plugs that new x into the appropriate spline function (either the function for values above 500 or the one for values below 500), then it takes that result and multiplies it by the appropriate coefficient and from that point treats it like any other model term. How do I get these spline functions?

(Note: I realize that a log-linked gamma GLM may not be appropriate for the data set provided. I am not asking about how or when to fit GLMs. I am providing that set as an example for reproducibility purposes.)

• I'd suggest, if at all possible, to avoid including code that deletes all variables (rm(list=ls())), especially not without any warning. Someone may copy-paste your code into an open session of R where they have some variables already (but none called x,y,df or spline1) and miss that your code wipes out their work. Is it kinda dumb for them to do that? Yes. But it's still polite to let them decide when to delete their own variables. May 16 '14 at 3:18
• It would serve them right for blindly copy-pasting code... Mar 3 '20 at 13:50

You can reverse-engineer the spline formulae without having to go into the R code. It suffices to know that

• A spline is a piecewise polynomial function.

• Polynomials of degree $d$ are determined by their values at $d+1$ points.

• The coefficients of a polynomial can be obtained via linear regression.

Thus, you only have to create $d+1$ points spaced between each pair of successive knots (including the implicit endpoints of the data range), predict the spline values, and regress the prediction against the powers of $x$ up to $x^d$. There will be a separate formula for each spline basis element within each such knot "bin." For instance, in the example below there are three internal knots (for four knot bins) and cubic splines ($d=3$) were used, resulting in $4\times 4=16$ cubic polynomials, each with $d+1=4$ coefficients. Because relatively high powers of $x$ are involved, it is imperative to preserve all the precision in the coefficients. As you might imagine, the full formula for any spline basis element can get pretty long!

As I mentioned quite a while ago, being able to use the output of one program as the input of another (without manual intervention, which can introduce irreproducible errors) is a useful statistical communication skill. This question provides a nice example of how that principle applies: instead of copying those $64$ sixteen-digit coefficients manually, we can hack together a way to convert the splines computed by R into formulas that Excel can understand. All we need do is extract the spline coefficients from R as described above, have it reformat them into Excel-like formulas, and copy and paste those into Excel.

This method will work with any statistical software, even undocumented proprietary software whose source code is unavailable.

Here is an example taken from the question, but modified to have knots at three internal points ($200, 500, 800$) as well as at the endpoints $(1, 1000)$. The plots show R's version followed by Excel's rendering. Very little customization was performed in either environment (apart from specifying colors in R to match Excel's default colors approximately).

(The vertical gray gridlines in the R version show where the internal knots are.)

Here is the full R code. It's an unsophisticated hack, relying entirely on the paste function to accomplish the string manipulation. (A better way would be to create a formula template and fill it in using string matching and substitution commands.)

#
# Create and display a spline basis.
#
x <- 1:1000
n <- ns(x, knots=c(200, 500, 800))

colors <- c("Orange", "Gray", "tomato2", "deepskyblue3")
plot(range(x), range(n), type="n", main="R Version",
xlab="x", ylab="Spline value")
for (k in attr(n, "knots")) abline(v=k, col="Gray", lty=2)
for (j in 1:ncol(n)) {
lines(x, n[,j], col=colors[j], lwd=2)
}
#
# Export this basis in Excel-readable format.
#
ns.formula <- function(n, ref="A1") {
ref.p <- paste("I(", ref, sep="")
knots <- sort(c(attr(n, "Boundary.knots"), attr(n, "knots")))
d <- attr(n, "degree")
f <- sapply(2:length(knots), function(i) {
s.pre <- paste("IF(AND(", knots[i-1], "<=", ref, ", ", ref, "<", knots[i], "), ",
sep="")
x <- seq(knots[i-1], knots[i], length.out=d+1)
y <- predict(n, x)
apply(y, 2, function(z) {
s.f <- paste("z ~ x+", paste("I(x", 2:d, sep="^", collapse=")+"), ")", sep="")
f <- as.formula(s.f)
b.hat <- coef(lm(f))
s <- paste(c(b.hat[1],
sapply(1:d, function(j) paste(b.hat[j+1], "*", ref, "^", j, sep=""))),
collapse=" + ")
paste(s.pre, s, ", 0)", sep="")
})
})
apply(f, 1, function(s) paste(s, collapse=" + "))
}
ns.formula(n) # Each line of this output is one basis formula: paste into Excel


The first spline output formula (out of the four produced here) is

"IF(AND(1<=A1, A1<200), -1.26037447288906e-08 + 3.78112341937071e-08*A1^1 + -3.78112341940948e-08*A1^2 + 1.26037447313669e-08*A1^3, 0) + IF(AND(200<=A1, A1<500), 0.278894459758071 + -0.00418337927419299*A1^1 + 2.08792741929417e-05*A1^2 + -2.22580643138594e-08*A1^3, 0) + IF(AND(500<=A1, A1<800), -5.28222778473101 + 0.0291833541927414*A1^1 + -4.58541927409268e-05*A1^2 + 2.22309136420529e-08*A1^3, 0) + IF(AND(800<=A1, A1<1000), 12.500000000002 + -0.0375000000000067*A1^1 + 3.75000000000076e-05*A1^2 + -1.25000000000028e-08*A1^3, 0)"


For this to work in Excel, all you need do is remove the surrounding quotation marks and prefix it with an "=" sign. (With a bit more effort you could have R write a file which, when imported by Excel, contains copies of these formulas in all the right places.) Paste it into a formula box and then drag that cell around until "A1" references the first $x$ value where the spline is to be computed. Copy and paste (or drag and drop) that cell to compute values for other cells. I filled cells B2:E:102 with these formulas, referencing $x$ values in cells A2:A102.

• ns.formula.. do you think in R?! Seriously though your method looks very useful but it seems ironic to have to hack a hack to get these parameters. Would be very useful to output a table.. Dec 16 '14 at 15:44
• This might be a stupid question: but is it 4 splines you are plotting, or 4 basis of one spline? Jan 28 '19 at 16:38
• @Erosennin I depends on what you mean by "one spline." These four curves are a basis for a spline that is piecewise cubic in four intervals and continuously second differentiable at the three points where those intervals meet, as described by the three bullet points that introduce my answer.
– whuber
Jan 28 '19 at 17:37
• Thanks! I did not mean to be nitpicking, It just looks as there are four splines (from the answer), and not four curves that are a basis. Again, I’m just here trying to understand... Jan 28 '19 at 17:44
• @Erosennin No problem. Maybe this will help: the "spline" is whatever linear combination of these four curves is determined by the regression fitting process. Another way to put it: the spline consists of a vector space of curves that can be created by taking linear combinations of these four curves.
– whuber
Jan 28 '19 at 17:59

You may find it easier to use the truncated power basis for cubic regression splines, using the R rms package. Once you fit the model you can retrieve the algebraic representation of the fitted spline function using the Function or latex functions in rms.

• Thank you. I actually read your response here stats.stackexchange.com/questions/67607/… before posting. I guess I just need a better grasp of what I can do with rms.
– Eric
May 15 '14 at 17:36
• The documentation for Function() does not really say what it does. In my case (see details on Rpubs rpubs.com/EmilOWK/rms_splines), I get function(x = NA) {-2863.7787+245.72672* x-0.1391794*pmax(x-10.9,0)^3+0.27835881*pmax(x-50.5,0)^3-0.1391794*pmax(x-90.1,0)^3 } <environment: 0x556156e80db8> The -2863.7787 value is the first coef in the model, the 245.72672 the second, and the last coef -873.0223 is not seen in the equation anywhere. The same applies to the output of latex(). May 2 '19 at 2:06
• Function works with Glm() when you use rcs as the spline function. The output is rephrasing the spline in simplest form by writing as if the linear tail restrictions are not there (but they are) as detailed in my RMS course notes. May 3 '19 at 0:59

> rm(list=ls())
> set.seed(1066)
> x<- 1:1000
> y<- rep(0,1000)
> y[1:500]<- pmax(x[1:500]+(runif(500)-.5)*67*500/pmax(x[1:500],100),0.01)
> y[501:1000]<-500+x[501:1000]^1.05*(runif(500)-.5)/7.5
> df<-as.data.frame(cbind(x,y))
> library(splines)
>


Now I will show you how to predict (the response) for x=12 in two different ways: First using the predict function (the easy way!)

> new.dat=data.frame(x=12)
> predict(spline1,new.dat,type="response")
1
68.78721


The 2nd way is based on the model matrix directly. Note I used exp since the link function used is log.

> m=model.matrix( ~ ns(df\$x,knots=c(500)))
> prd=exp(coefficients(spline1) %*% t(m))
> prd[12]
[1] 68.78721


Note that in above I extracted the 12th element, since that correspond to x=12. If you want to predict for an x outside the training set, then simply you can again use the predict function. Lets say we want to find the predicted response value for x=1100 then

> predict(spline1, newdata=data.frame(x=1100),type="response")
1
366.3483

• Thank you for your response! But, I'm still confused :/. I'm not sure I know what to do with this matrix. For example, if I had x = 12, then predict says y = 68.78721, but looking up 12 from that matrix I get 0.016816392. The original intercept and coefficient for x<500 is 4.174603 and 3.830416, respectively. exp(4.174603+3.8304116*0.016816392) <> 68.78721. Plus, how would I get values for x if x was not in the training set?
– Eric
May 15 '14 at 15:05