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43 votes
3 answers

Relation between confidence interval and testing statistical hypothesis for t-test

It is well known that confidence intervals and testing statistical hypothesis are strongly related. My questions is focused on comparison of means for two groups based on a numerical variable. Let's ...
Lan's user avatar
  • 1,371
236 votes
4 answers

How to interpret a QQ plot

I am working with a small dataset (21 observations) and have the following normal QQ plot in R: Seeing that the plot does not support normality, what could I infer about the underlying distribution? ...
JohnK's user avatar
  • 19.4k
200 votes
8 answers

What intuitive explanation is there for the central limit theorem?

In several different contexts we invoke the central limit theorem to justify whatever statistical method we want to adopt (e.g., approximate the binomial distribution by a normal distribution). I ...
user avatar
146 votes
9 answers

What is the difference between linear regression on y with x and x with y?

The Pearson correlation coefficient of x and y is the same, whether you compute pearson(x, y) or pearson(y, x). This suggests that doing a linear regression of y given x or x given y should be the ...
user9097's user avatar
  • 3,203
133 votes
7 answers

How to choose between t-test or non-parametric test e.g. Wilcoxon in small samples

Certain hypotheses can be tested using Student's t-test (maybe using Welch's correction for unequal variances in the two-sample case), or by a non-parametric test like the Wilcoxon paired signed rank ...
Silverfish's user avatar
  • 22.1k
117 votes
4 answers

Assessing approximate distribution of data based on a histogram

Suppose I want to see whether my data is exponential based on a histogram (i.e. skewed to the right). Depending on how I group or bin the data, I can get wildly different histograms. One set of ...
guestoeijreor's user avatar
77 votes
5 answers

How can adding a 2nd IV make the 1st IV significant?

I have what is probably a simple question, but it is baffling me right now, so I am hoping you can help me out. I have a least squares regression model, with one independent variable and one ...
EvKohl's user avatar
  • 1,150
63 votes
3 answers

Interpretation of log transformed predictor and/or response

I'm wondering if it makes a difference in interpretation whether only the dependent, both the dependent and independent, or only the independent variables are log transformed. Consider the case of <...
upabove's user avatar
  • 2,807
176 votes
10 answers

Bottom to top explanation of the Mahalanobis distance?

I'm studying pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahalanobis distance. The books give sort of intuitive explanations, but still ...
jjepsuomi's user avatar
  • 5,617
79 votes
6 answers

Optimization when Cost Function Slow to Evaluate

Gradient descent and many other methods are useful for finding local minima in cost functions. They can be efficient when the cost function can be evaluated quickly at each point, whether numerically ...
Jared Becksfort's user avatar
64 votes
5 answers

What should I do when my neural network doesn't generalize well?

I'm training a neural network and the training loss decreases, but the validation loss doesn't, or it decreases much less than what I would expect, based on references or experiments with very similar ...
DeltaIV's user avatar
  • 16.9k
56 votes
3 answers

Consider the sum of $n$ uniform distributions on $[0,1]$, or $Z_n$. Why does the cusp in the PDF of $Z_n$ disappear for $n \geq 3$?

I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the ...
tetragrammaton's user avatar
277 votes
11 answers

How would you explain covariance to someone who understands only the mean?

...assuming that I'm able to augment their knowledge about variance in an intuitive fashion ( Understanding "variance" intuitively ) or by saying: It's the average distance of the data ...
PhD's user avatar
  • 14.2k
32 votes
1 answer

How do you deal with "nested" variables in a regression model?

Consider a statistical problem where you have a response variable that you want to describe conditional on an explanatory ...
Ben's user avatar
  • 112k
553 votes
15 answers

What is the intuition behind beta distribution?

Disclaimer: I'm not a statistician but a software engineer. Most of my knowledge in statistics comes from self-education, thus I still have many gaps in understanding concepts that may seem trivial ...
ffriend's user avatar
  • 9,720
261 votes
15 answers

What are the differences between Factor Analysis and Principal Component Analysis?

It seems that a number of the statistical packages that I use wrap these two concepts together. However, I'm wondering if there are different assumptions or data 'formalities' that must be true to use ...
Brandon Bertelsen's user avatar
30 votes
3 answers

Is my model any good, based on the diagnostic metric ($R^2$/ AUC/ accuracy/ RMSE etc.) value?

I've fitted my model and am trying to understand whether it's any good. I've calculated the recommended metrics to assess it ($R^2$/ AUC / accuracy / prediction error / etc) but do not know how to ...
mkt's user avatar
  • 15.9k
557 votes
11 answers

What is the difference between test set and validation set?

I found this confusing when I use the neural network toolbox in Matlab. It divided the raw data set into three parts: training set validation set test set I notice in many training or learning ...
xiaohan2012's user avatar
  • 7,099
95 votes
4 answers

What is an "uninformative prior"? Can we ever have one with truly no information?

Inspired by a comment from this question: What do we consider "uninformative" in a prior - and what information is still contained in a supposedly uninformative prior? I generally see the prior in ...
Fomite's user avatar
  • 21.9k
73 votes
5 answers

How small a quantity should be added to x to avoid taking the log of zero?

I have analysed my data as they are. Now I want to look at my analyses after taking the log of all variables. Many variables contain many zeros. Therefore I add a small quantity to avoid taking the ...
miura's user avatar
  • 3,584
57 votes
2 answers

Is it unusual for the MEAN to outperform ARIMA?

I recently applied a range of forecasting methods (MEAN, RWF, ETS, ARIMA and MLPs) and found that MEAN did surprisingly well. (MEAN: where all future predictions are predicted as been equal to the ...
Andy T's user avatar
  • 1,114
19 votes
1 answer

Question on how to normalize regression coefficient

Not sure if normalize is the correct word to use here, but I will try my best to illustrate what I am trying to ask. The estimator used here is least squares. Suppose you have $y=\beta_0+\beta_1x_1$, ...
Saber CN's user avatar
  • 799
60 votes
4 answers

Box-Cox like transformation for independent variables?

Is there a Box-Cox like transformation for independent variables? That is, a transformation that optimizes the $x$ variable so that the y~f(x) will make a more ...
Tal Galili's user avatar
  • 20.8k
53 votes
4 answers

Does the sign of scores or of loadings in PCA or FA have a meaning? May I reverse the sign?

I performed principal component analysis (PCA) with R using two different functions (prcomp and princomp) and observed that the ...
user1320502's user avatar
36 votes
3 answers

How to tell the difference between linear and non-linear regression models?

I was reading the following link on non linear regression SAS Non Linear. My understanding from reading the first section "Nonlinear Regression vs. Linear Regression" was that the equation below is ...
mHelpMe's user avatar
  • 657
81 votes
0 answers

How can a regression be significant yet all predictors be non-significant? [duplicate]

My multiple regression analysis model has a statistically significant F value however all beta values are statistically non-significant. All the regression assumptions are met. No multicollinearity ...
Serene's user avatar
  • 811
36 votes
2 answers

Interpretation of simple predictions to odds ratios in logistic regression

I'm somewhat new to using logistic regression, and a bit confused by a discrepancy between my interpretations of the following values which I thought would be the same: exponentiated beta values ...
mike's user avatar
  • 847
95 votes
1 answer

When to use an offset in a Poisson regression? [duplicate]

Does anybody know why offset in a Poisson regression is used? What do you achieve by this?
MarkDollar's user avatar
  • 5,785
59 votes
3 answers

How to select a clustering method? How to validate a cluster solution (to warrant the method choice)?

One of the biggest issue with cluster analysis is that we may happen to have to derive different conclusion when base on different clustering methods used (including different linkage methods in ...
Learner's user avatar
  • 909
142 votes
3 answers

Removal of statistically significant intercept term increases $R^2$ in linear model

In a simple linear model with a single explanatory variable, $\alpha_i = \beta_0 + \beta_1 \delta_i + \epsilon_i$ I find that removing the intercept term improves the fit greatly (value of $R^2$ ...
Ernest A's user avatar
  • 2,252
193 votes
5 answers

Training on the full dataset after cross-validation?

TL:DR: Is it ever a good idea to train an ML model on all the data available before shipping it to production? Put another way, is it ever ok to train on all data available and not check if the model ...
Amelio Vazquez-Reina's user avatar
88 votes
9 answers

Why is it possible to get significant F statistic (p<.001) but non-significant regressor t-tests?

In a multiple linear regression, why is it possible to have a highly significant F statistic (p<.001) but have very high p-values on all the regressor's t tests? In my model, there are 10 ...
Ηλίας's user avatar
  • 1,529
440 votes
5 answers

How to understand the drawbacks of K-means

K-means is a widely used method in cluster analysis. In my understanding, this method does NOT require ANY assumptions, i.e., give me a dataset and a pre-specified number of clusters, k, and I just ...
KevinKim's user avatar
  • 6,709
278 votes
2 answers

Interpretation of R's lm() output

The help pages in R assume I know what those numbers mean, but I don't. I'm trying to really intuitively understand every number here. I will just post the output and comment on what I found out. ...
Alexander Engelhardt's user avatar
135 votes
4 answers

Nested cross validation for model selection

How can one use nested cross validation for model selection? From what I read online, nested CV works as follows: There is the inner CV loop, where we may conduct a grid search (e.g. running K-fold ...
Amelio Vazquez-Reina's user avatar
75 votes
4 answers

Why is sample standard deviation a biased estimator of $\sigma$?

According to the Wikipedia article on unbiased estimation of standard deviation the sample SD $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}$$ is a biased estimator of the SD of the ...
Dav Weps's user avatar
  • 759
313 votes
10 answers

What's the difference between a confidence interval and a credible interval?

Joris and Srikant's exchange here got me wondering (again) if my internal explanations for the difference between confidence intervals and credible intervals were the correct ones. How you would ...
Matt Parker's user avatar
  • 5,837
236 votes
14 answers

How should I transform non-negative data including zeros?

If I have highly skewed positive data I often take logs. But what should I do with highly skewed non-negative data that include zeros? I have seen two transformations used: $\log(x+1)$ which has the ...
Rob Hyndman's user avatar
  • 54.5k
85 votes
5 answers

Central limit theorem for sample medians

If I calculate the median of a sufficiently large number of observations drawn from the same distribution, does the central limit theorem state that the distribution of medians will approximate a ...
user1728853's user avatar
  • 1,047
60 votes
4 answers

Confidence interval for Bernoulli sampling

I have a random sample of Bernoulli random variables $X_1 ... X_N$, where $X_i$ are i.i.d. r.v. and $P(X_i = 1) = p$, and $p$ is an unknown parameter. Obviously, one can find an estimate for $p$: $\...
user avatar
108 votes
5 answers

Loadings vs eigenvectors in PCA: when to use one or another?

In principal component analysis (PCA), we get eigenvectors (unit vectors) and eigenvalues. Now, let us define loadings as $$\text{Loadings} = \text{Eigenvectors} \cdot \sqrt{\text{Eigenvalues}}.$$ I ...
user2696565's user avatar
  • 1,379
72 votes
9 answers

Which pseudo-$R^2$ measure is the one to report for logistic regression (Cox & Snell or Nagelkerke)?

I have SPSS output for a logistic regression model. The output reports two measures for the model fit, Cox & Snell and ...
Henrik's user avatar
  • 13.9k
176 votes
2 answers

Deriving the conditional distributions of a multivariate normal distribution

We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning $\boldsymbol\mu$ and ${\boldsymbol Y}$ into $$\boldsymbol\mu = \begin{bmatrix} \...
Flying pig's user avatar
  • 6,079
104 votes
12 answers

What, precisely, is a confidence interval?

I know roughly and informally what a confidence interval is. However, I can't seem to wrap my head around one rather important detail: According to Wikipedia: A confidence interval does not ...
dsimcha's user avatar
  • 8,449
147 votes
6 answers

Correlations with unordered categorical variables

I have a dataframe with many observations and many variables. Some of them are categorical (unordered) and the others are numerical. I'm looking for associations between these variables. I've been ...
Clément F's user avatar
  • 1,767
84 votes
5 answers

Intuition on the Kullback–Leibler (KL) Divergence

I have learned about the intuition behind the KL Divergence as how much a model distribution function differs from the theoretical/true distribution of the data. The source I am reading goes on to say ...
cgo's user avatar
  • 8,547
64 votes
5 answers

Correlations between continuous and categorical (nominal) variables

I would like to find the correlation between a continuous (dependent variable) and a categorical (nominal: gender, independent variable) variable. Continuous data is not normally distributed. Before, ...
Md. Ferdous Wahid's user avatar
229 votes
18 answers

Intuitive explanation for dividing by $n-1$ when calculating standard deviation?

I was asked today in class why you divide the sum of square error by $n-1$ instead of with $n$, when calculating the standard deviation. I said I am not going to answer it in class (since I didn't ...
Tal Galili's user avatar
  • 20.8k
52 votes
1 answer

Alternatives to one-way ANOVA for heteroskedastic data

I have data from 3 groups of algae biomass ($A$, $B$, $C$) which contain unequal sample sizes ($n_A=15$, $n_B=13$, $n_C=12$) and I would like compare if these groups are from the same population. One-...
Rick L.'s user avatar
  • 521
32 votes
3 answers

I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of ...
kjetil b halvorsen's user avatar

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