## Building a regression model is easy and informative

Running an experiment is very time-consuming. I am always surprised that people put so much effort into gathering the data and then spend so little effort analyzing it.

The Computer Language Benchmarks Game looks like a fun benchmark; it compares the performance of 27 languages using various toy benchmarks (they could not be said to be representative of real programs). And, yes, lots of boxplots and tables of numbers; great eye-candy, but what do they all mean?

The authors, like good experimentalists, make all their data available. So, what analysis should they have done?

A regression model is the obvious choice and the following three lines of R (four lines if you could the blank line) build one, providing lots of interesting performance information:

cl=read.csv("Computer-Language_u64q.csv.bz2", as.is=TRUE) cl_mod=glm(log(cpu.s.) ~ name+lang, data=cl) summary(cl_mod) |

The following is a cut down version of the output from the call to `summary`

, which summarizes the model built by the call to `glm`

.

Estimate Std. Error t value Pr(>|t|) (Intercept) 1.299246 0.176825 7.348 2.28e-13 *** namechameneosredux 0.499162 0.149960 3.329 0.000878 *** namefannkuchredux 1.407449 0.111391 12.635 < 2e-16 *** namefasta 0.002456 0.106468 0.023 0.981595 namemeteor -2.083929 0.150525 -13.844 < 2e-16 *** langclojure 1.209892 0.208456 5.804 6.79e-09 *** langcsharpcore 0.524843 0.185627 2.827 0.004708 ** langdart 1.039288 0.248837 4.177 3.00e-05 *** langgcc -0.297268 0.187818 -1.583 0.113531 langocaml -0.892398 0.232203 -3.843 0.000123 *** Null deviance: 29610 on 6283 degrees of freedom Residual deviance: 22120 on 6238 degrees of freedom

What do all these numbers mean?

We start with `glm`

's first argument, which is a specification of the regression model we are trying to fit: `log(cpu.s.) ~ name+lang`

`cpu.s.`

is cpu time, `name`

is the name of the program and `lang`

is the language. I found these by looking at the column names in the data file. There are other columns in the data, but I am running in quick & simple mode. As a first stab, I though cpu time would depend on the program and language. Why take the `log`

of the cpu time? Well, the model fitted using cpu time was very poor; the values range over several orders of magnitude and logarithms are a way of compressing this range (and the fitted model was much better).

The model fitted is:

, or

Plugging in some numbers, to predict the cpu time used by say the program `chameneosredux`

written in the language `clojure`

, we get: (values taken from the first column of numbers above).

This model assumes there is no interaction between program and language. In practice some languages might perform better/worse on some programs. Changing the first argument of `glm`

to: `log(cpu.s.) ~ name*lang`

, adds an interaction term, which does produce a better fitting model (but it's too complicated for a short blog post; another option is to build a mixed-model by using `lmer`

from the `lme4`

package).

We can compare the relative cpu time used by different languages. The multiplication factor for `clojure`

is , while for `ocaml`

it is . So `clojure`

consumes 8.2 times as much cpu time as `ocaml`

.

How accurate are these values, from the fitted regression model?

The second column of numbers in the `summary`

output lists the estimated standard deviation of the values in the first column. So the `clojure`

value is actually , i.e., between 2.2 and 4.9 (the multiplication by 1.96 is used to give a 95% confidence interval); the `ocaml`

values are , between 0.3 and 0.6.

The fourth column of numbers is the p-value for the fitted parameter. A value of lower than 0.05 is a common criteria, so there are question marks over the fit for the program `fasta`

and language `gcc`

. In fact many of the compiled languages have high p-values, perhaps they ran so fast that a large percentage of start-up/close-down time got included in their numbers. Something for the people running the benchmark to investigate.

Isn't it easy to get interesting numbers by building a regression model? It took me 10 minutes, ok I spend a lot of time fitting models. After spending many hours/days gathering data, spending a little more time learning to build simple regression models is well worth the effort.

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