167

The title of this question suggests a fundamental misunderstanding. The most basic idea of correlation is "as one variable increases, does the other variable increase (positive correlation), decrease (negative correlation), or stay the same (no correlation)" with a scale such that perfect positive correlation is +1, no correlation is 0, and perfect negative ...


117

It depends on what sense of a correlation you want. When you run the prototypical Pearson's product moment correlation, you get a measure of the strength of association and you get a test of the significance of that association. More typically however, the significance test and the measure of effect size differ. Significance tests: Continuous vs. ...


65

You're right on both counts. See Frank Harrell's page here for a long list of problems with binning continuous variables. If you use a few bins you throw away a lot of information in the predictors; if you use many you tend to fit wiggles in what should be a smooth, if not linear, relationship, & use up a lot of degrees of freedom. Generally better to ...


34

As a fact of history, regression and ANOVA developed separately, and, due in part to tradition, are still often taught separately. In addition, people often think of ANOVA as appropriate for designed experiments (i.e., the manipulation of a variable / random assignment) and regression as appropriate for observational research (e.g., downloading data from a ...


25

The standard Cauchy distribution is derived from the ratio of two independent Normal Distributions. If $X \sim N(0,1)$, and $Y \sim N(0,1)$, then $\tfrac{X}{Y} \sim \operatorname{Cauchy}(0,1)$. The Cauchy distribution is important in physics (where it’s known as the Lorentz distribution) because it’s the solution to the differential equation describing ...


23

Is there a sharp discontinuity at your thresholds? For instance, suppose you have two patients A and B with values 3.9 and 4.1, and another two patients C and D with values 6.7 and 6.9. Is the difference in the likelihood for cancer between A and B much larger than the corresponding difference between C and D? If yes, then discretizing makes sense. If not,...


22

That's an interesting question. My research group has been using the distribution you refer to for some years in our publicly available bioinformatics software. As far as I know, the distribution does not have a name and there is no literature on it. While the paper by Chandra et al (2012) cited by Aksakal is closely related, the distribution they consider ...


19

Aggregation is substantively meaningful (whether or not the researcher is aware of that). One should bin data, including independent variables, based on the data itself when one wants: To hemorrhage statistical power. To bias measures of association. A literature starting, I believe, with Ghelke and Biehl (1934—definitely worth a read, and suggestive of ...


19

Look at this paper: Chandra, Nimai Kumar, and Dilip Roy. A continuous version of the negative binomial distribution. Statistica 72, no. 1 (2012): 81. It's defined in the paper as the survival function, which is a natural approach since neg binomial was introduced in reliability analysis: $$S_r(x)=\begin{cases}q^x & \text{for}\ r=1 \\ \sum_{k=0}^{r-1}...


19

In addition to its usefulness in physics, the Cauchy distribution is commonly used in models in finance to represent deviations in returns from the predictive model. The reason for this is that practitioners in finance are wary of using models that have light-tailed distributions (e.g., the normal distribution) on their returns, and they generally prefer to ...


17

Nominal vs Interval The most classic "correlation" measure between a nominal and an interval ("numeric") variable is Eta, also called correlation ratio, and equal to the root R-square of the one-way ANOVA (with p-value = that of the ANOVA). Eta can be seen as a symmetric association measure, like correlation, because Eta of ANOVA (with the nominal as ...


16

Looks like you're also looking for an answer from a predictive standpoint, so I put together a short demonstration of two approaches in R Binning a variable into equal sized factors. Natural cubic splines. Below, I've given the code for a function that will compare the two methods automatically for any given true signal function test_cuts_vs_splines <- ...


15

So you've been told you need an appropriate distance measure. Here are some leads: Clustering mixed data A generalized Mahalanobis distance for mixed data Estimating the Mahalanobis distance from mixed continuous and discrete data Generalization of the Mahalanobis distance in the mixed case Distance functions for categorical and mixed variables ...


15

I've seen the following cheatsheet linked before: https://stats.idre.ucla.edu/other/mult-pkg/whatstat/ It may be useful to you. It even has links to specific R libraries.


15

It is a slight exaggeration to say that binning should be avoided at all costs, but it is certainly the case that binning introduces bin choices that introduce some arbitrariness to the analysis. With modern statistical methods it is generally not necessary to engage in binning, since anything that can be done on discretized "binned" data can generally be ...


14

The answer is exactly 0 in theory and approximately 0 in practice. Let $X$ be a continuous random variable. Then $Y=X_i-X_j$ is also continuous. If $P(Y=0)=0$ then the probability of two observations of $X$ being equal is $0$, since $$P(X_i=X_j)=P(X_i-X_j=0)=P(Y=0)=0.$$ If $P(Y=0)>0$ then the probability of doublets is greater than $0$. To see that $P(...


13

I am a little confused; your title says "correlation" but your post refers to t-tests. A t-test is a test of central location - more specifically, is the mean of one set of data different from the mean of another set? Correlation, on the other hand, shows the relationship between two variables. There are a variety of correlation measures, it seems that ...


13

Categorical solution Treating the values as categorical loses the crucial information about relative sizes. A standard method to overcome this is ordered logistic regression. In effect, this method "knows" that $A\lt B\lt \cdots \lt J\lt \ldots$ and, using observed relationships with regressors (such as size) fits (somewhat arbitrary) values to each ...


13

Assuming proportional hazards (as in a Cox model) and the hazard ratio for a 1 mg increase in nicotine smoked a day is 1.02, then this tells you that persons smoking 11 mgs were 1.02 as likely to die in the monitored time period than persons smoking 10 mgs. The same applies to 12 vs 11 mgs etc. If the units of your continuous covariable are too small for ...


13

The result can be proven with a picture: the visible gray areas show that a uniform distribution cannot be decomposed as a sum of two independent identically distributed variables. Notation Let $X$ and $Y$ be iid such that $X+Y$ has a uniform distribution on $[0,1]$. This means that for all $0\le a \le b \le 1$, $$\Pr(a < X+Y \le b) = b-a.$$ The ...


13

Probabilities are models for the relative frequencies of observations. If an event $A$ is observed to have occurred $N_A$ times on $N$ trials, then its relative frequency is $$\text{relative frequency of }(A) = \frac{N_A}{N}$$ and it is generally believed that the numerical value of the above ratio is a close approximation to $P(A)$ when $N$ is "large" ...


12

Let $(\Omega,\mathscr{F},P)$ be the underlying probability space. We say that a measurable function $X:\Omega\to\mathbb{R}$ is an absolutely continuous random variable if the probability measure $\mu_X$ over $(\mathbb{R},\mathscr{B})$ defined by $\mu_X(B)=P\{X\in B\}$, known as the distribution of $X$, is dominated by Lebesgue measure $\lambda$, in the sense ...


11

That is not a necessary result, but it is certainly plausible. If you turn a quantitive predictor into a single categorical predictor you lose a lot of information; with the categorical predictor you only know whether an observation is below or above a certain threshold (e.g. the mean or median), while with a quantitative predictor you also know how much ...


11

There is, as far as I know, no taxonomy of variables that captures all the contrasts that might be important for some theoretical or practical purpose, even for statistics alone. If such a taxonomy existed, it would probably be too complicated to be widely acceptable. It is best to focus on examples rather than give numerous definitions. Number of days is ...


11

A part of this answer that I've learned since asking is that not binning and binning seeks to answer two slightly different questions - What is the incremental change in the data? and What is the difference between the lowest and the highest?. Not binning says "this is a quantification of the trend seen in the data" and binning says "I don't have enough ...


11

If you're talking about an interaction in a general linear model (e.g., ANCOVA), and if your categorical moderator has a reasonably small number of levels, you can plot separate regression lines for each level of the moderator. If you want these on the same plot, superimpose them, code by color or line type, and provide a legend. One of your plot's axes will ...


11

This seems like a job for survival analysis, like Cox proportional hazards analysis or possibly some parametric survival model. Think about this problem in reverse from the way you're explaining it: what are the predictor variables associated with earlier distances to quitting? Quitting is the event. The distance covered might be considered equivalent to ...


11

Actually nobody says that such event is impossible. Probability equal to zero is not the same as impossibility (check here, here and here). Probability that $X=x$ is equal to zero for continuous variables because chance of such event happening is infinitely small since there is an infinite number of real numbers. Also from purely mathematical point of view ...


10

Discrete Data can only take certain values. Example: the number of students in a class (you can't have half a student). Continuous Data is data that can take any value (within a range) Examples: A person's height: could be any value (within the range of human heights), not just certain fixed heights, Time in a race: you could even measure it to ...


10

I tried finding a proof without considering characteristic functions. Excess kurtosis does the trick. Here's the two-line answer: $\text{Kurt}(U) = \text{Kurt}(X + Y) = \text{Kurt}(X) / 2$ since $X$ and $Y$ are iid. Then $\text{Kurt}(U) = -1.2$ implies $\text{Kurt}(X) = -2.4$ which is a contradiction as $\text{Kurt}(X) \geq -2$ for any random variable. ...


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