8

A statistic is a function that maps from the set of outcomes of the observable values to a real number. Thus, with $n$ data points, a statistic will be a function $s: \mathbb{R}^n\rightarrow \mathbb{R}$ as in your second form. However, it is also possible to view the statistic in its random sense by taking the appropriate composition of function with the ...


6

What can we say about the two versions? You are directly modelling the probability distributions for the two conversion rates. You can use the posterior to answer questions about the two conversion rates. Such questions might be... What is the probability the new version has a larger conversion rate? What is the probability the new version has a smaller ...


6

You are right that the documentation is wrong. Note that p values are defined somewhat differently from what you write. They do not measure the probability of a decision, such as the decision to reject the null or to fail to reject the null. They measure the probability of test statistics. Whether or not to reject a null hypothesis is a subsequent decision ...


5

Well, I don't know about "publication ready", but you can try using the effects package in R to obtain predictor effects plots. See https://cran.r-project.org/web/packages/effects/vignettes/predictor-effects-gallery.pdf for details and also the R code below. Example 1 [Categorical by Continuous Interaction]: Let's say you fit the model below in R and are ...


5

Let's look at three possible models: \begin{align} \mathrm{logit}(y) &= \beta_0 + \beta_1\cdot \mathrm{hours} + \beta_2\cdot \mathrm{course} + \beta_{3}\cdot \mathrm{hours}\times\mathrm{course}\tag{1} \\ \mathrm{logit}(y) &= \beta_0 + \beta_1\cdot \mathrm{hours} + \beta_2\cdot \mathrm{course}\tag{2}\\ \mathrm{logit}(y) &= \beta_0 + \beta_1\cdot \...


5

In models with nonlinear link functions there is indeed a difference in the interpretation of the regression coefficients in GEEs and mixed-effects models. In short, GEEs give you the more usual interpretation of comparing groups of subjects. E.g., for dichotomous outcomes and the logit link you get the log-odds ratio between the group of males and the ...


4

In short: in such an small-sample non-randomized trial you can't tell apart the effects of year and type of surgical procedure. You could check what has changed with years and argue that the main change has been type of procedure, and this argument could be sound but it would not be an statistical argument. In general, it's difficult to infer causation just ...


4

Positive coefficients somehow indicate a positive effect, but they don't simply turn into percentages. There is a transformation. Let's say your model is $\log y = b_0+b_1x_1$; this means $y=e^{b_0+b_1x_1}=A_0e^{b_1x_1}$. So, dummy or not, if $x_1$ increases by $1$ unit, $y$ increases by $e^{b_1}$, i.e. if $b_1=0.2$, $y$ increases by $e^{0.2}\approx 1.22$, i....


4

"Sign-arbitrariness" is merely an artifact of how we represent the PCA results. There is no arbitrariness to the PCA itself: the eigenspaces it works with are perfectly well defined. Issues (1) and (3) are advantages of PCA, because they allow one to use subject-matter knowledge and the objectives of the analysis appropriately. Referring to this as ...


4

The question is a good one: when including an angular variable $w$ among the explanatory variables in any regression model by incorporating its sine and cosine, how does one interpret the two estimated coefficients? Let's write this mathematically just to make things perfectly clear. Suppose you model a response $Y$ in the form $$Y = X\beta + \alpha_1 \...


4

When you say the data are "stationary", do you mean all of the following: There is no evidence of a temporal trend in the values of flow over time; There is no evidence of increased variability in the values of flow over time; There is no evidence of a temporal trend in the values of ec over time; There is no evidence of increased variability in the ...


4

As per my comment, once you fit your model, you can extract the values of the predictors included in the model using the model.matrix() function: require(stats) require(splines) require(graphics) fm1 <- lm(weight ~ bs(height, df = 5), data = women) summary(fm1) model.matrix(fm1) The R output produced by model.matrix() for your model is as follows: ...


4

Assuming that the x axis is number of components and the y axis is variance explained, if the relationship really was linear, it would say that your PCA is not working at all, as each PC is extracting the same amount of variance. That could happen if all your variables were orthogonal to each other. Your graph is quite far from that. I don't think the lack ...


4

what you did - you created confidence intervals under assumption that chicken weights are drawn from normal disrtibution (with value range $(-\infty, \infty)$) - in fact these can be drawn from other disrtibution with $\mathbb{R_+}$ support e.g. erlang or chi distribution, but when sample size is $> 50$ we can assume that mean has normal disrtibution - so ...


3

That advice from your supervisor seems strange to me, the usual approach would be to use all predictors in one model. You can find more discussion in a good answer: Choosing the proper statistical approach for glm But it is also useful, as a way of understanding the data better, to fit multiple models, including also models with only one predictor. Here is ...


3

With dunn.test you have an argument altp which sets how the p-value will be expressed. If in function call you set altp=TRUE, then the p values will be expressed in alternative format. Test default is to express p-value = P(Z ≥ |z|), and reject Ho if p ≤ α/2. So what you describe sounds like normal behaviour. You still can change it - if the altp option is ...


3

What is being tested Your (1) indicates that you tested for 0 difference in population medians and your (2) indicates that you tested for the population median of pair-differences being 0. Strictly, a signed rank test is not testing what either of your interpretations say. The actual population quantity being considered is the pseudomedian of population ...


3

It could be considered as probability of output being 1 only in case if classifier is calibrated. For example, in weather forecasting it's important to have reliable rain probability estimation. Let's say your model says that there is 80% that there will be rain tomorrow. Your classifier will be considered calibrated only when you observe predictions with 80%...


3

The event $(X > 10)$ is the union of the disjoint events $(X=11), (X=12),(X=13), \cdots$ and so $P(X>10)$ is the sum of the probabilities of these events. The hard way of calculating $P(X>10)$ is to sum the (geometric) series to arrive at $0.95^{10}$. The easier way to get to the same answer is by musing on the fact that the only way that the event $...


3

All these are incorrect because of misunderstandings about what probabilities, error bars and confidence bands mean. I recommend reading a good textbook about them, and this paper specifically about interpreting error bars: Error bars in experimental biology. No, this is wrong for multiple reasons. (i) The line does not appear to go exactly to zero, just ...


3

There's no indication that there's any mistake here. PCA is a dimension reduction technique, so it will find orthogonal vectors. If you have many correlated variables, it's not surprising that many of them will be strongly loaded onto a single dimension. It might be worth your time to read our most-upvoted thread on CV, which has many great explanations of ...


3

I would just (first of all) work on logarithmic scale and back-transform the confidence limits obtained on that scale. That way you're assured of positive limits. Going full Bayes on this is an answer of wide appeal, but as you're asking this question I am not clear that "learn a whole new approach to statistics" is likely to be practical immediately for ...


2

When the procedure you have used to calculate a confidence interval gives an interval including impossible values, that is an indication of problems with the method. In your case, you have used a normal (central limit theorem-based) CI with so few observations that the approximation is invalid. You can test that easily in R, say: We have plotted the ...


2

(Percent correct)/(Total count) is usually termed the Correct Classification Rate. This is not one of the pseudo-R-squared indicators, and it's generally considered an inferior way of assessing model fit because it simplifies so much; it doesn't take into account the differences in predicted probability from observation to observation. The pseudo-R-squared ...


2

This is an old question, but for anyone who stumbles across it in the future, intuitively the 2SLS estimate of $\beta_1$ is $\alpha_1$ from the "reduced form" regression $$y = \alpha_0 + \alpha_1 z_1 + \mathbf{Z}\mathbf{\alpha} + u$$ divided by $\pi_1$ from the "first stage" regression $$x_1 = \pi_0 + \pi_1z_1 + \mathbf{Z}\mathbf{\pi} + v$$ So if the ...


2

You can see the degree of freedom as the number of observations minus the number of necessary relations among these observations. By exemple if you have $n$ sample of independant normal distribution observations $X_1,\dots,X_n$. The random variable $\sum_{i=1}^n (X_i-\overline{X}_n)^2\sim \mathcal{X}^2_{n-1}$, where $\overline{X}_n = \frac{1}{n}\sum_{i=1}^n ...


2

There is nothing inherently wrong with measuring goodness-of-fit in a regression model with other monotonic transforms of the coefficient of determination, and indeed, there are several natural transformations that have useful interpretations. Before getting to these, it is useful to note the desirable mathematical properties of the coefficient of ...


2

Splitting the dataset on gender is not a good idea. You lose the possibility to model and interpret the interactions (and you lose statistical power.) More concretely, splitting the file on gender is the same as including all interactions with gender in the model (well, in addition it gets more variance estimates.) The full details can be found in my answer ...


2

Typically in machine learning (or statistics), we don't think of features/covariates as random variables. Or at least, we often don't care about the random nature of our features. We just want to model $P(y | X)$, where $y$ is the outcome of interest, and $X$ is our set of covariates. We care about the random nature of $y$ given $X$, but we assume $X$ is ...


2

Quoting my other answer: Naive Bayes algorithm makes the "naive" assumption, that the features are (conditionally) independent, so by definition of independence $$ p(x_1, x_2, \dots, x_k \mid y) = \prod_{i=1}^k p(x_i | y) $$ [...] the estimate of conditional probability would be accurate if the assumption of independence holds (and in real ...


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