134

First of all, we should understand what the R software is doing when no intercept is included in the model. Recall that the usual computation of $R^2$ when an intercept is present is $$ R^2 = \frac{\sum_i (\hat y_i - \bar y)^2}{\sum_i (y_i - \bar y)^2} = 1 - \frac{\sum_i (y_i - \hat y_i)^2}{\sum_i (y_i - \bar y)^2} \>. $$ The first equality only occurs ...


126

A Dialog Between a Teacher and a Thoughtful Student Humbly submitted in the belief that not enough crayons have been used so far in this thread. A brief illustrated synopsis appears at the end. Student: What does a p-value mean? A lot of people seem to agree it's the chance we will "see a sample mean greater than or equal to" a statistic or it's "the ...


108

The problem with t-SNE is that it does not preserve distances nor density. It only to some extent preserves nearest-neighbors. The difference is subtle, but affects any density- or distance based algorithm. To see this effect, simply generate a multivariate Gaussian distribution. If you visualize this, you will have a ball that is dense and gets much less ...


81

To me "1 in 80 deaths..." is by far the clearer statement. The denominator in your "1 in 80" is the set of all death events and that statement makes it explicit. There's ambiguity in the "1 in 80 people..." formulation. You really mean "1 in 80 people who dies..." but the statement can just as easily be interpreted as "1 in 80 people now alive..." or ...


80

Imagine your job is to forecast the number of Americans that will die from various causes next year. A reasonable place to start your analysis might be the National Vital Statistics Data final death data for 2014. The assumption is that 2017 might look roughly like 2014. You'll find that approximately 2,626,000 Americans died in 2014: 614,000 died of heart ...


75

What you have done is logistic regression. This can be done in basically any statistical software, and the output will be similar (at least in content, albeit the presentation may differ). There is a guide to logistic regression with R on UCLA's excellent statistics help website. If you are unfamiliar with this, my answer here: difference between logit ...


67

First of all, my first intuitive thought was: "S2 can only be the same as S1 if the traffic death rate stays constant, possibly over decades" - which certainly wouldn't have been a good assumption in the last so many decades. This already hints that one difficulty lies with implicit/unspoken temporal assumptions. I'd say your statements have the form 1 in ...


56

My detailed answer is below, but the general (i.e. real) answer to this kind of question is: 1) experiment, screw around, look at the data, you can't break the computer no matter what you do, so . . . experiment; or 2) RTFM. Here is some R code which replicates the problem identified in this question, more or less: # This program written in response to a ...


53

Parameter estimates, like a sample mean or an OLS regression coefficient, are sample statistics that we use to draw inferences about the corresponding population parameters. The population parameters are what we really care about, but because we don't have access to the whole population (usually assumed to be infinite), we must use this approach instead. ...


50

It seems self-evident to me that $$ \exp(\beta_0 + \beta_1x) \neq\frac{\exp(\beta_0 + \beta_1x)}{1+\exp(\beta_0 + \beta_1x)} $$ unless $\exp(\beta_0 + \beta_1x)=0$. So, I'm less clear about what the confusion might be. What I can say is that the left hand side (LHS) of the (not) equals sign is the odds of being undernourished, whereas the RHS is the ...


49

The answer depends on whether you are assuming the symmetric or asymmetric dirichlet distribution (or, more technically, whether the base measure is uniform). Unless something else is specified, most implementations of LDA assume the distribution is symmetric. For the symmetric distribution, a high alpha-value means that each document is likely to contain a ...


43

Charlie provides a nice, correct explanation. The Statistical Computing site at UCLA has some further examples: http://www.ats.ucla.edu/stat/sas/faq/sas_interpret_log.htm , and http://www.ats.ucla.edu/stat/mult_pkg/faq/general/log_transformed_regression.htm Just to complement Charlie's answer, below are specific interpretations of your examples. As always, ...


43

Note that the Shapiro-Wilk is a powerful test of normality. The best approach is really to have a good idea of how sensitive any procedure you want to use is to various kinds of non-normality (how badly non-normal does it have to be in that way for it to affect your inference more than you can accept). An informal approach for looking at the plots would ...


43

The Kullback-Leibler Divergence is not a metric proper, since it is not symmetric and also, it does not satisfy the triangle inequality. So the "roles" played by the two distributions are different, and it is important to distribute these roles according to the real-world phenomenon under study. When we write (the OP has calculated the expression using ...


43

It depends on whether you are describing or predicting. "1 in 80 people will die in a car accident" is a prediction. Of all the people alive today, some time within their remaining lifetime, one in 80 will die that way. "1 in 80 deaths are caused by a car accident" is a description. Of all the people who died in a given period (e.g. the time span of a ...


40

Your fitted model with lme() can be expressed as $y_{ij} = \alpha_0 + \alpha_1 x_j + \delta_{0i} + \delta_{1i} x_j + \epsilon_{ij}$ where $y_{ij}$ is the score of $i$th employee at $x_j$ weeks, $\alpha_0$ and $\alpha_1$ are the fixed intercept and slope respectively, $\delta_{0i}$ and $\delta_{1i}$ are the random intercept and slope, and $\epsilon_{ij}$ is ...


39

Causal theory offers another explanation for how two variables could be unconditionally independent yet conditionally dependent. I am not an expert on causal theory and am grateful for any criticism that will correct any misguidance below. To illustrate, I will use directed acyclic graphs (DAG). In these graphs, edges ($-$) between variables represent ...


36

I think that you need to remember that ARIMA models are atheoretic models, so the usual approach to interpreting estimated regression coefficients does not really carry over to ARIMA modelling. In order to interpret (or understand) estimated ARIMA models, one would do well to be cognizant of the different features displayed by a number of common ARIMA ...


35

I would like to provide a somewhat dissenting opinion to the well argued (+1) and highly upvoted answer by @ErichSchubert. Erich does not recommend clustering on the t-SNE output, and shows some toy examples where it can be misleading. His suggestion is to apply clustering to the original data instead. use t-SNE for visualization (and try different ...


34

Reason for rotation. Rotations are done for the sake of interpretation of the extracted factors in factor analysis (or components in PCA, if you venture to use PCA as a factor analytic technique). You are right when you describe your understanding. Rotation is done in the pursuit of some structure of the loading matrix, which may be called simple structure. ...


31

I assume that PreferA = 1 when one prefered A and 0 otherwise and that ControlFALSE = 1 when treated and 0 when control. The odds of preffering A when a person did not do so previously and did not receive a treatment (ControlFALSE=0 and PreferA=0) is $\exp(3.135)= 23$, i.e. there are 23 such persons who prefer A for every such person that prefers B. So A is ...


31

Interpretation of deep models is still challenging. Your post only mentions CNNs for computer vision applications, but (deep or shallow) feed-forward networks and recurrent networks remain challenging to understand. Even in the case of CNNs which have obvious "feature detector" structures, such as edges and orientation of pixel patches, it's not completely ...


29

1) I would argue that decision trees are not as interpretable as people make them out to be. They look interpretable, since each node is a simple, binary decision. The problem is that as you go down the tree, each node is conditional on every node above it. If your tree is only four or five levels deep, it's still not too difficult to convert one terminal ...


28

I don't think the title of your question accurately captures what you're asking for. The question of how to interpret the parameters in a GLM is very broad because the GLM is a very broad class of models. Recall that a GLM models a response variable $y$ that is assumed to follow a known distribution from the exponential family, and that we have chosen an ...


26

I haven't seen RMSLE before, but I'm assuming it's $\sqrt{ \frac{1}{N} \sum_{i=1}^N (\log(x_i) - \log(y_i))^2 }$. Thus exponentiating it won't give you RMSE, it'll give you $e^\sqrt{ \frac{1}{N} \sum_{i=1}^N (\log(x_i) - \log(y_i))^2 } \ne \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - y_i)^2}$. If we take the log of both sides, we get the RMSLE versus $\frac{1}{2}...


26

You are right. Technically, it is any value. However, when I teach this I usually tell people that you are getting the effect of a one unit change in $X_j$ when all other variables are held at their respective means. I believe this is a common way to explain it that is not specific to me. I usually go on to mention that if you don't have any ...


26

Problems with the chart: It implies refugees are more likely than other groups of people to commit acts of terror. Why not frame it in terms of migrants in general? And what about acts of terror committed by a country's own citizens? How does it define a refugee? The comparative groups don't make sense. If we are going to look a killings why not compare it ...


25

The log-linked gamma GLM specification is identical to exponential regression: $$E[y \vert x,z] = \exp \left( \alpha + \beta \cdot x +\gamma \cdot z \right)=\hat y$$ This means that $E[y \vert x=0,z=0]=\exp(\alpha)$. That's not a very meaningful value (unless you centered your variables to be be mean zero beforehand). There are at least three way to ...


25

It will almost never be meaningful to use the no intercept model in logistic regression. The intercept parameter $\beta_0$ is modelling the marginal distribution of the response $Y$, so using $\beta_0=0$ is tantamont to assuming that $P(Y=1)=0.5$, marginally. Do you really know that? If that is untrue, you cannot trust any inference from the no intercept ...


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