I want to compare the curve fits of three models by r-squared values. I ran models using the nls and drc packages. It appears, though, that neither of those packages calculate r-squared values; they give "residual std error" and "residual sum of squares" though.
Can these two be used to compare model fits?
This is really a statistics question, rather than a coding question: consider posting on stats.stackexchange.com; you're likely to get a better answer.
RSQ is not really meaningful for non-linear regression. This is why summary.nls(...) does not provide it. See this post for an explanation.
There is a common, and understandable, tendency to hope for a single statistic that allows one to assess which of a set of models better fits a dataset. Unfortunately, it doesn't work that way. Here are some things to consider.
Generally, the best model is the one that has a mechanistic underpinning. Do your models reflect some physical process, or are you just trying a bunch of mathematical equations and hoping for the best? The former approach almost always leads to better models.
You should consider how the models will be used. Will you be interpolating (e.g. estimating y|x within the range of your dataset), or will you be extrapolating (estimating y|x outside the range of your data)? Some models yield a fit that provides relatively accurate estimates slightly outside the dataset range, and others completely fall apart.
Sometimes the appropriate modeling technique is suggested by the type of data you have. For example, if you have data that counts something, then y is likely to be poisson distributed and a generalized linear model (glm) in the poisson family is indicated. If your data is binary (e.g. only two possible outcomes, success or failure), then a binomial glm is indicated (so-called logistic regression).
The key underlying assumption of least squares techniques is that the error in y is normally distributed with mean 0 and constant variance. We can test this after doing the fit by looking at a plot of standardized residuals vs. y, and by looking at a Normal Q-Q plot of the residuals. If the residuals plot shows scatter increasing or decreasing with y then the model in not a good one. If the Normal Q-Q plot is not close to a straight line, then the residuals are not normally distributed and probably a different model is indicated.
Sometimes certain data points have high leverage with a given model, meaning that the fit is unduly influenced by those points. If this is a problem you will see it in a leverage plot. This indicates a weak model.
For a given model, it may be the case that not all of the parameters are significantly different from 0 (e.g., p-value of the coefficient > 0.05). If this is the case, you need to explore the model without those parameters. With nls, this often implies a completely different model.
Assuming that your model passes the tests above, it is reasonable to look at the F-statistic for the fit. This is essentially the ratio of SSR/SSE corrected for the dof in the regression (R) and the residuals (E). A model with more parameters will generally have smaller residual SS, but that does not make it a better model. The F-statistic accounts for this in that models with more parameters will have larger regression dof and smaller residual dof, making the F-statistic smaller.
Finally, having considered the items above, you can consider the residual standard error. Generally, all other things being equal, smaller residual standard error is better. Trouble is, all other things are never equal. This is why I would recommend looking at RSE last.
Related
I am trying to develop Cox PH model with time-varying covariates in R. I use coxph function from survival package. There was not any trouble during estimation process, though coefficient value of one covariates is too large, in particular, 2.5e+32.
I can't guess what is reason of this problem and how to tackle it. This variable is nonstationary and proportional assumption is violated. Does either of this facts may cause such a big value of coefficient?
More information could help framing your problem.
Anyway, I doubt non-proportionality is to blame. It would imply that you have some outliers heavily biasing your coefficient beyond reasonable expectations. You could give this a quick look by plotting the output of cox.zph.
Another possible explanation is that this rather depends on the unit of measure you used to define your covariate. Can the magnitude of the coefficient be meaningfully interpreted? If so, you could simply re-scale/standardise/log-transform that covariate to obtain a 'more manageable' coefficient (if this is theoretically appropriate).
This could also be due to the so called 'complete separation', which has been discussed here and here.
Working with a dataset of ~200 observations and a number of variables. Unfortunately, none of the variables are distributed normally. If it possible to extract a data subset where at least one desired variable will be distributed normally? Want to do some statistics after (at least logistic regression).
Any help will be much appreciated,
Phil
If there are just a few observations that skew the distribution of individual variables, and no other reasons speaking against using a particular method (such as logistic regression) on your data, you might want to study the nature of "weird" observations before deciding on which analysis method to use eventually.
I would:
carry out the desired regression analysis (e.g. logistic regression), and as it's always required, carry out residual analysis (Q-Q Normal plot, Tukey-Anscombe plot, Leverage plot, also see here) to check the model assumptions. See whether the residuals are normally distributed (the normal distribution of model residuals is the actual assumption in linear regression, not that each variable is normally distributed, of course you might have e.g. bimodally distributed data if there are differences between groups), see if there are observations which could be regarded as outliers, study them (see e.g. here), and if possible remove them from the final dataset before re-fitting the linear model without outliers.
However, you always have to state which observations were removed, and on what grounds. Maybe the outliers can be explained as errors in data collection?
The issue of whether it's a good idea to remove outliers, or a better idea to use robust methods was discussed here.
as suggested by GuedesBF, you may want to find a test or model method which has no assumption of normality.
Before modelling anything or removing any data, I would always plot the data by treatment / outcome groups, and inspect the presence of missing values. After quickly looking at your dataset, it seems that quite some variables have high levels of missingness, and your variable 15 has a lot of zeros. This can be quite problematic for e.g. linear regression.
Understanding and describing your data in a model-free way (with clever plots, e.g. using ggplot2 and multiple aesthetics) is much better than fitting a model and interpreting p-values when violating model assumptions.
A good start to get an overview of all data, their distribution and pairwise correlation (and if you don't have more than around 20 variables) is to use the psych library and pairs.panels.
dat <- read.delim("~/Downloads/dput.txt", header = F)
library(psych)
psych::pairs.panels(dat[,1:12])
psych::pairs.panels(dat[,13:23])
You can then quickly see the distribution of each variable, and the presence of correlations among each pair of variables. You can tune arguments of that function to use different correlation methods, and different displays. Happy exploratory data analysis :)
I'm modelling multiple linear regression. I used the bptest function to test for heteroscedasticity. The result was significant at less than 0.05.
How can I resolve the issue of heteroscedasticity?
Try using a different type of linear regression
Ordinary Least Squares (OLS) for homoscedasticity.
Weighted Least Squares (WLS) for heteroscedasticity without correlated errors.
Generalized Least Squares (GLS) for heteroscedasticity with correlated errors.
Welcome to SO, Arun.
Personally, I don't think heteroskedasticity is something you "solve". Rather, it's something you need to allow for in your model.
You haven't given us any of your data, so let's assume that the variance of your residuals increases with the magnitude of your predictor. Typically a simplistic approach to handling it is to transform the data so that the variance is constant. One way of doing this might be to log-transform your data. That might give you a more constant variance. But it also transforms your model. Your errors are no longer IID.
Alternatively, you might have two groups of observarions that you want to compare with a t-test, bit the variance in one group is larger than in the other. That's a different sot of heteroskedasticity. There are variants of the standard "pooled variance" t-test that might handle that.
I realise this isn't an answer to your question in the conventional sense. I would have made it a comment, but I knew before I started that I'd need more words than a comment would let me have.
I had a dataset for which I needed to provide a linear regression model that represents diameter as a function of length.Data which has length in first column and diameter in second looked like:
0.455,0.365
0.44,0.365
I carried out the required operations on the given dataset in R,and plotted the regression line for the data
I am just confused about what to conclude from the parameters(slope=0.8154, y intercept:-0.019413, correlation coefficient:0.98 ). Can I conclude anything other than line is a good fit. I am new to statistics. Any help would be appreciated.
Slope 0.8154 informs you that each unit increase for lenght causes increase of diamater in 0.8154*unit. Intercept -0.019413 is probably statistically insignificant in this case. To verify that you have to look at t-statistics for example.
On this page you can find nice course with visualizations about simple linear regression and other statistical methods answering your questions.
From the parameters slope and intercept, you cannot conclude if the line is a good fit. The correlation coefficient says that they depend highly and that a straight line could fit your model. However, from the p-values for the slope and intercept, you can conclude if your fit is good. If they are small (say below 0.05) you can conclude that the fit is pretty good.
What functions do you use in R to fit a curve to your data and test how well that curve fits? What results are considered good?
Just the first part of that question can fill entire books. Just some quick choices:
lm() for standard linear models
glm() for generalised linear models (eg for logistic regression)
rlm() from package MASS for robust linear models
lmrob() from package robustbase for robust linear models
loess() for non-linear / non-parametric models
Then there are domain-specific models as e.g. time series, micro-econometrics, mixed-effects and much more. Several of the Task Views as e.g. Econometrics discuss this in more detail. As for goodness of fit, that is also something one can spend easily an entire book discussing.
The workhorses of canonical curve fitting in R are lm(), glm() and nls(). To me, goodness-of-fit is a subproblem in the larger problem of model selection. Infact, using goodness-of-fit incorrectly (e.g., via stepwise regression) can give rise to seriously misspecified model (see Harrell's book on "Regression Modeling Strategies"). Rather than discussing the issue from scratch, I recommend Harrell's book for lm and glm. Venables and Ripley's bible is terse, but still worth a reading. "Extending the Linear Model with R" by Faraway is comprehensive and readable. nls is not covered in these sources, but "Nonlinear Regression with R" by Ritz & Streibig fills the gap and is very hands-on.
The nls() function (http://sekhon.berkeley.edu/stats/html/nls.html) is pretty standard for nonlinear least-squares curve fitting. Chi squared (the sum of the squared residuals) is the metric that is optimized in that case, but it is not normalized so you can't readily use it to determine how good the fit is. The main thing you should ensure is that your residuals are normally distributed. Unfortunately I'm not sure of an automated way to do that.
The Quick R site has a reasonable good summary of basic functions used for fitting models and testing the fits, along with sample R code:
http://www.statmethods.net/stats/regression.html
The main thing you should ensure is
that your residuals are normally
distributed. Unfortunately I'm not
sure of an automated way to do that.
qqnorm() could probably be modified to find the correlation between the sample quantiles and the theoretical quantiles. Essentially, this would just be a numerical interpretation of the normal quantile plot. Perhaps providing several values of the correlation coefficient for different ranges of quantiles could be useful. For example, if the correlation coefficient is close to 1 for the middle 97% of the data and much lower at the tails, this tells us the distribution of residuals is approximately normal, with some funniness going on in the tails.
Best to keep simple, and see if linear methods work "well enuff". You can judge your goodness of fit GENERALLY by looking at the R squared AND F statistic, together, never separate. Adding variables to your model that have no bearing on your dependant variable can increase R2, so you must also consider F statistic.
You should also compare your model to other nested, or more simpler, models. Do this using log liklihood ratio test, so long as dependant variables are the same.
Jarque–Bera test is good for testing the normality of the residual distribution.