Bias, in a statistical framework, means that an estimate of a parameter has an expected value that is not equal to the actual parameter value. There is often a tradeoff between bias and variance - low variance estimators may be more biased than high variance ones.

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R packages that work with biased samples

I'm working with a biased sample of web users. I'm only able to track responses of users who have navigated my site in a certain way, and I'd like to run an analysis to determine how certain factors ...
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51 views

Does BIAS equal to MEAN ERROR

Bias is defined as an average of all errors (without abs) and this is, IMO, what I want. However, I have been asked to give MEAN ERROR. Is this the same than bias and is it wrong to call bias as mean ...
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Forecasting and auto-correlation [duplicate]

I'm reading this chapter forecasting principles and practise from a forecasting book. The author has explained a linear regression model. Now this linear regression model will definitely have some ...
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40 views

statistical handling of lab values below limit of quantitation (BLQ)

There were several samples BLQ because of the lower limit of quantitation (LLQ) of the method, e.g. 5 ng/ml or less. Using the statistical program PRISM6 I marked these values together with the ...
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32 views

Comparison between normal glm and glm.nb regression with quadratic term?

Let's say I have a function to simulate data for negative binomial regression: ...
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73 views

How do instrumental variables address selection bias?

I'm wondering how an instrumental variable addresses selection bias in regression. Here's the example I'm chewing on: In Mostly Harmless Econometrics, the authors discuss and IV regression relating ...
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10 views

Calculating Bias from Cox PH model

I am using COX PH model to fit a lifetime data set and estimating parameters. By the by I am also simulating data and trying to find out estimates of parameters. Now I want to calculate bias and ...
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19 views

handling multicollinearity by backwards regression and omitted variable bias

Suppose I try to estimate a production function as follows: $logY=b_0+b1*logX_1+b_2*logX_2+b_{11}*(logX_1)^2+b_{22}*(logX_2)^2+b_{12}*(logX_1)*(logX_2)+u$, where $Y$ is the output, $X_1$, $X_2$ are ...
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Estimating the probability of causation based on finding a correlation, including experimental details

Say there is a hypothesis that A causes B (A -> B), and some likelihood that the hypothesis is correct (AB1%). Now, an experiment is run that claims to find a correlation between A and B. What I ...
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68 views

Why use d-prime instead of percent correct?

In signal detection theory, people often use $d'$ to assess performance. Apart from the fact that $d'$ is in $z$ units (units of measurement transformed to standard deviation units, i.e., $z$ scores), ...
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19 views

Finding parameter bias under omitted variable, with variance covariance notation

Dear CrossValidated community, Can anyone help me to prove the bias in a given parameter of a regression when there is omitted variable? I know to do it using matrices and matrix algebra. For ...
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47 views

Proof that omitted variable bias may lead to endogeneity

I am looking for a proof that omitted variable bias (OVB) in OLS regression may lead to endogeneity. I have found many examples here and out there on how to prove that a given parameter $b_{j}$ (where ...
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61 views

Is the bootstrap estimate of the mean biased when a single extreme value is in the sample?

My sample includes $n$ random observations, while $n-1$ of these observations are in the range (0-1) there is also one observation that gets very high value. For example, a sample of prices where ...
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28 views

Optimism bias - estimates of prediction error

The book Elements of Statistical Learning (available in PDF online) discusses the optimisim bias (7.21, page 229). It states that the optimism bias is the difference between the training error and the ...
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9 views

RTM and study design issue

I was asked to estimate "regression to the mean" (RTM) error for the purpose of measuring program efficacy for reducing spending among the highest spending individuals. Given the constraints of the ...
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3answers
435 views

Intuitive reasoning behind biased ML estimators

I have a confusion on biased Maximum Likelihood estimators. The mathematics of the whole concept is pretty clear to me but I cannot figure out the intuitive reasoning behind it. Given a certain ...
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23 views

Selection Bias with Groups

I have conducted a pre-post test study with a set of 20 participants. There were two independent variables and therefore 4 groups of 5 participants each. The experiment took place in a classroom and ...
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1answer
30 views

Longitudinal panel dataset: Consequences of missing values

I am analyzing a longitudinal panel dataset using OLS. The data spans around 40 years, but for some variables data was unavailable for certain categories. In most cases, the data for given category ...
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2answers
111 views

Bias and Variance - Errors in R example

I am trying to better understand the bias and variance trade-off, and tried to create a R example. It attempts to calculate the bias and variance of smoothing splines with different parameters. ...
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40 views

Bias and variance estimation with boostrap

The Wikipedia article about Jacknife estimation of the bias and variance of an estimator $\theta$ includes the following formulas: Variance of $\theta$: $ \operatorname {Var}(\theta )=\sigma ...
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40 views

Reason for not shrink the bias term

For linear model, $y=\beta_0+x*\beta+\varepsilon$, the shrinkage term is always like $P(\beta) $. What's the reason we do not shrink the bias term $\beta_0$? Comparatively, should we shrink the bias ...
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63 views

How to correct sample size bias in logistic regression?

I'm working on a logistic regression with N = 42, which is too small following Hosmer's and Lemeshow's recommendations of at least N = 100. How will this affect my logistic regression and is there a ...
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21 views

Adjusting regression for correlated errors-in-variables

I have a set of data points $\{x_i\}$. These data points are grouped so that (say) $i\in\{1,2,3\}$ is group $A$, $i\in\{4,5,6,7\}$ is group $B$, etc. I would like to test the null hypothesis of no ...
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44 views

Weird bootstrap bias for Predictor Importance (MeanDecreaseAccuracy) in Random Forests

Below, using R, I: 1. Create a data set with a bunch of factors. All of them are predictors and 'y' is the dependent variable. 2. I run a classification Random Forests for y with predictor importance. ...
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57 views

Estimating Bias in a Linear Regression Model

Is there any way to estimate the bias of the estimate of the betas in a linear regression model when the actual beta values are unknown? The well known Mean Square Error (MSE) criterion is used to ...
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1answer
54 views

Asymptotics of OLS coefficients for unequal variance RHS variables

This seems to be a very general question about the bias OLS produces for RHS variables with unequal variance, but I was not able to find an explicit solution anywhere. Suppose we have realizations of ...
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Is the sign test appropriate to test whether values in one distribution are generally lower than the other?

Generally, the sign test is used to test the hypothesis that the difference median is zero between two continuous distributions (Sign test). I am trying to understand whether it can be adapted to ...
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85 views

Are bias weights essential in the output layer, if one wants a universal function approximator (or non-linearly separable problem solver)?

I am learning about ELM (Extreme Learning Machines) and it appears to have no bias weights at the output layer. Besides that, just to clarify, the kind of ELM I am refering to are topologically no ...
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21 views

Does measurement error include effects of moderator variables?

I am not very sure how to interpret measurement error correctly: as a constant, as a bias, or as a moderator factor? Is it presumed that measurement error includes moderator effects? Let it be in ...
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344 views

Can a magician bias the result of a fair die?

Some time ago, Persi Diaconis was discussing probability and made the point that the level of information a person possessed about an event played a role in his accessing probabilities. To illustrate ...
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1answer
75 views

What could cause different conversion rates for the same content in separate split tests?

We have an events based tracking system for our website, with split testing built-in and we are using ABBA for the calculations. The problem comes up when we are doing consecutive split tests. For ...
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1answer
97 views

Estimators, sufficiency, consistency, and bias

A random variable is said to have the Pareto distribution with parameters $\alpha$ and $\beta$, $P(\alpha, \beta)$, if its cumulative distribution function is given by $$F(x)= 1 - ...
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58 views

What are the pitfalls of using subjective ratings for correlation and causal explanation?

Take a study that collects subjective guesstimates as a proxy for some variable, like the quality of service. In a simplest model, we then regress the monthly sales on these guesstimates. How to ...
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60 views

Quality of a model and the bias-variance tradeoff

Take linear regression as the example, given one specific data set $D_1=\{(x_1,y_1),...(x_n,y_n)\}$, we could train a model with one specific parameter estimate $\hat\theta_1$, if we do the training ...
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172 views

If the dependent variable is standardized by age and sex, does it still make sense to include these as controls in a multivariate regression?

Suppose I have a dependent variable that is a test score, $score^*_{i}$. It has been standardised for both age and gender. I want to measure the effect that a binary variable, say Adoption, has on ...
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41 views

calculate the bias and variance of the MLE in an exponential family

I'm trying to calculate the bias and variance of $\hat{\lambda}_{MLE}$ where $X \sim exponential(\lambda)$. Secondly, does it attain the CRLB? Attempt: $$f_x(X) = \lambda e^{-\lambda x}$$ ...
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47 views

Why does one bootstrap sample help reduce bias in a biased variance estimator?

I am looking a bit on the effect of bootstrap on bias. I came by the following example in the estimation of variance (when dividing by n instead of n-1). Can someone explain to me why the bias when ...
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33 views

How to calculate differential item functioning when factor structure differs between groups?

I want to test for differential item functioning on a self-report measure between two groups (i.e., one with a disease one without). Differential item functioning refers to a measurement bias wherein ...
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33 views

OLS unbiased for all sample sizes

I am pretty positive that OLS regression produces unbiased estimates for all sample sizes, even though the variance about those estimates might become very large when sample sizes become small (e.g. ...
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47 views

Does Bayesian data analysis take into account estimates' bias?

For example, the standard deviation is known to have bias that depends on the number of samples observed. If I wanted to do Bayesian inference on the SD of samples from two populations, and have ...
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48 views

Bias-variance tradeoff — bias or variance effect

I understand that supervised learning is associated with an error that can be split between bias and variance: $ MSE = b^2 + var $ What does bias and variance account for intuitively ? I ...
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1answer
165 views

How do I check for bias of an estimator?

I need to check if an estimator $\hat\theta$ for the parameter $\theta$ is biased. Theory says I should compare the expected value of $\hat\theta$ versus the expected value of $\theta$. I assume the ...
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1answer
51 views

self-selection bias due to nonresponse?

I have administrative data from the whole population of new doctorates, in a given year, from my region. We have also survey data from a sample of this same population (where the whole population was ...
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1answer
131 views

Pearson correlation between discrete variable that's mostly 0 and a standard normal variable

Suppose I want to estimate the correlation between $X\sim N(0,1)$ and $Y$, where $Y \in \{-1,0,1\}$ and is equal to zero for 99 per cent of the sample. Sample size is 10 million. What are the ...
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36 views

Using expectation to detect bias

I was going through Penn State's online notes and noticed this expression: $ v^2 = \frac{1}{n} \sum_{i=1}^n (y_i - \bar{y})^2$ In the line below it they stated that the $E[v^2] = (1 - ...
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76 views

Detrending Discrete Data

I am trying to detrend some discrete data and I am having difficulty finding a model to describe the trend. There is a number of discrete data points and there is a linear error being introduced with ...
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65 views

Can logistic regression estimates suffering from subsample abuse be salvaged?

Suppose we have some logistic regression modelling problem; $f(X) = Y$, where $Y$ is binary and $X$ is a vector of normally distributed variables. In industry it is sometimes the case that ...
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Proper Sampling - can I collect a two-group sample this way without issues?

I need to collect a two group sample for a comparison analysis (perhaps using logistic regression). The population that I need to extract a sample from is all firms from country A with activities in ...
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Interpretation of quantile regression when high quantile estimates are lower

Suppose I estimate a multivariable quantile regression $Q_Y(\tau | X) = \alpha(\tau) + \beta(\tau)X + \epsilon(\tau)$. Note that $X$ is a vector of independent variables. Suppose I then 'plug in' my ...
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What is the difference between the concept and treatment of measurement error in psychometry and in statistics?

There is some confusion with respect to the measurement error. What is the definition in statistics and definition in psychometry ? The statistics does not seem to recognize the measurement error ...