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Your question says "Which method is better?". Better what for? If you want to predict GPA you might want to use both variables. If your question is about the relation between IQ and GPA, then you have no reason to add age to the Model. Hence, it depends on your research question what Model suits better. One point that appears unmentioned, is that not only ...


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What this would do is answer drastically different questions. Multiple regressions of one independent variable will give you an understand of the target variable varies with each output of each variable A regression with multiple independent variables would give you coefficient estimates that let you know how the target variable varies for a given change ...


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Difference 1 Below is a sketch of a hypothetical relationship for GPA as function of age and IQ. Added to this are the fitted lines for the two different situations. If you add the effects of two regressions with one independent variable then you can see this as obtaining a relationship for the slope of GPA as function of IQ and the slope of GPA as ...


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The best practice is to convert to a column that's 1 if it's a male, 0 if it's a female. You may want to switch these, I think 1 for male is more common. Other encodings are used, for example -1 and 1. This influences the interpretation of the coefficients that are fitted with a regression model. Other than the interpretation and the actual value of the ...


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I think if you put an inverse-gamma prior on $\sigma^2$ then the posterior distribution of $\sigma^2$ will be an inverse-gamma (provided, of course, that you have normal prior on the coefficients). Please see page 54 of Bayesian Core for more details. HTH


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Good question because, although you have to notice a couple of things that I will mention below, it is a good point for many real-life applications. First of all, just to be precise, when you say “In building the model, I think it makes sense to try to capture the individual effects of, and interactions between, my predictors.”, you have to notice that when ...


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Feature or covariate are the terms that I use. Another that one might use is explanatory variable.


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To explain a little more. Multiple regression tests for the unique contribution of each predictor. So let's take your example and assume that IQ and age are correlated. If you run a regression with IQ only the contribution of IQ can be visualized like this (red part): But once you add age to the analysis, it looks something like that: As you can see the ...


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You can do that. It answers a different question. If you include both independent variables then the results for each are controlling for the other. If you do them separately then they are not.


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The predicted mean response at $x_i$ (the estimated conditional expectation of $y_i$, $E(y_i|x=x_i)$ would be of the form $\hat{\alpha} + \hat{\beta} x_i$. This is sometimes denoted as $\hat{y}_i$. In the example, $\hat{y}_i$ is the mean/expected number of ice cream sales at temperature $x_i$.


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As mentioned in the other answers, the existence of the random effects is irrelevant. As a matter of R code, your function does not include a*c. If that was intentional, be aware that a*b*c is automatically expanded to: a:b + a:c + b:c + a:b:c. That is, the term will be included. To deliberately exclude it, you should list the terms using the colon ...


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The coefficient for the interaction of a and b is interpreted the same way in a mixed effects model like this as it is for a regular linear model. It is the effect of each 1 unit increase in a*b. In including this term you assume that the effect of a on Y increases linearly with b. The interaction term reflects the slope of this linear increase. That is ...


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As you state the coefficient for a gives the expected increase in Y if you increase a by 1. Similarly for b. Suppose there is no interaction then if we simultaneously increase a and b by 1 then the expected increase in Y is simply the sum of the coefficients for a and b. If there is an interaction then the coefficient for a.b has to be added to the sum of a ...


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More than one response variable is known as multivariate regression. More than one predictor variable is known as multiple regression (compared to simple regression for a single variable).


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Your sample is tiny. In a single study, I wouldn't expect to find any significance at all unless the true effect was huge (and even then the effect would be estimated with little precision). If you set your significance level to be .1, then if there were no true effects in the data, you would expected to see significant findings 10% of the time. Indeed, you ...


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Let $\varepsilon \sim \mathcal{N}_n(0, \sigma^2 I_n)$, then $$ \mathbb{P}\big [ \Vert \hat{\beta}_n - \beta \Vert \geq \zeta \big ] = \mathbb{P}\big [ \sqrt{(\hat{\beta}_n - \beta )^T ( \hat{\beta}_n - \beta ) }\geq \zeta \big] $$ Note that $\hat{\beta}_n - \beta \sim \mathcal{N}(0 , \sigma^2 (X^T X)^{-1})$ Thus let $U_n = (\hat{\beta}_n - \beta )^T ( \...


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You wrote I'm picking variables with a p less or equal to 0.2 to include in the multivariate model, This is known as bivariate screening and it is a terrible method of building a model. This has been discussed here before or see Frank Harrell's book Regression Modelling Strategies. but when I include it in my model, which I have to because I specify ...


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Intuitively, our answer must consider some projection of the data onto the residual from the prediction of B: Q provides an orthonormal basis for B, so we must remove the components on Q. More rigorously, We can directly compute the MLE using the formula for linear estimator. We have that $$ \begin{bmatrix} A & B \end{bmatrix} \begin{bmatrix} \alpha \\ \...


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We have to be thinking about a model to answer your question so let's assume a linear model. For convenience, we'll use sums of squared deviations instead of variances; to translate for variances, divide through the sums of squares by $N - 1$. Let $Z = (z_1, ..., z_N)$ be your data; it has sum of squared deviations $\sum_{i = 1}^N (z_i - \bar{z})^2$. If you ...


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In a couple of words (the shorter the better!) when you add a variable to a model, if the added variable adds some explanatory power, then the addition increases the model fit (i.e. the capacity of the model as a whole of predicting the dependent variable in the sample where the model is estimated). However, bear in mind that adding more variables also ...


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I'm not familiar with the subject of the paper, so I don't have a motivation for the conversion, but from a pure mathematical perspective, the desired formulation can be reached via some algebraical tricks: $$\begin{align}D&=c_1+\beta_D\beta_UE+\epsilon_1=c_1+\epsilon_1+\beta_D\beta_U\frac{1+\beta_M^2}{1+\beta_M^2}E\\&=c_1+\epsilon_1+\frac{\beta_D\...


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You want to identify "variables that are most strongly related to the outcomes of complaints against practitioners in a profession," but not to predict future outcomes of complaints. Presumably, the idea is to generate hypotheses about factors that might be manipulated in future work to reduce undesirable outcomes. Cross-validation to choose a LASSO model, ...


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Your question is, IMHO, slightly off the point. In statistics book often a distinction is made between inference and prediction (e.g. in Harrell 2001 Regression Modeling Strategies, or in Shmueli 2010's paper To explain or to predict?). In your case, I would argue you are actually interested in using the data to form an hypothesis, i.e. explorative data ...


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The model will not be used to make predictions on future outcomes, but to make inferences about decisions during the time period. Having lots of hope, that I am not mistaken, I understand, that your goal is to make causal inferernce. This means you want to say "such and such decision caused different probability of outcome". Correct me if I am wrong. ...


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This post suggests that the computational challenges of standard dominance analysis (or "Shapley Regression") are well known as computational time grows exponentially with the number of predictors. Instead, a popular alternative is to perform "relative weights" analysis given that simulations suggest that this generally produces very similar results, but is ...


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Your reasoning is correct - as already commented by @user158565, a single model is generally better, as you are able to obtain a proper CI for the heterogeneity term. Another major difference is that the interaction model constrains the differences to that term, or the intercept, while the stratum-specific approach allows different effects for each of the ...


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The heteroscedasticity-robust standard error take into account the whole regressor matrix $X$, in the case of HC4 this is the following "sandwich" formula $HC4 = (X'X)^{-1} X' \text{diag} \Big[ \frac{e_i^2}{(1-h_{ii})^{\delta_i}}\Big] X(X'X)^{-1}$ with $\delta_i = \text{min} \Big\{4,\frac{nh_{ii}}{1-p}\Big\}$ Source here Since an interaction term is ...


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In your first model, the intercept is the expected value for group A when pred1, pred2, and pred3 are all 0. The coefficients for group B and group C are the expected difference from Group A for these other groups (at any value of the pred vars as you have no interactions). (Your example is not fully reproducible as you did not set seed. So I repeat with a ...


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I found this: https://www.originlab.com/doc/Origin-Help/Multi-Regression-Dialog So yes, we have regression lines for every independent variable.


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A multiple regression line is a line in a p+1 dimensional space, where p is the number of predictors (or independent variables). With p > 2 this will be hard to visualize, but we statisticians don't let that stop us. You could make a line relating each predictor to the DV, controlling for the other predictors, but you have to decide what levels of all the ...


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I would not expect this to be a problem, at all. The whole point of GDP/capita is that it is a way of representing GDP that makes sense across different values of the population - or in other words is "independent" (perhaps not in the strict statistical sense of the word, but to a "first approximation" [again not in the strict mathematical sense of the word])...


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OK, I'll post my comment as an answer. So: Why not categorize each person as lean/overweight/obese acording to his/her sex-and-age-specific norms and use this categorization as a predictor? mkt's answer is also fine, but I'd stick to categorization. It'll give you conclusions like "Being obese inrceases odds of hypertension by 50% compared to being lean". ...


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I'd recommend just using the continuous body fat values instead of binning them into groups. That allows you to avoid any problems with different classification thresholds, and also uses all the information you have (information is thrown out when you bin continuous data). Then you can do a logistic regression with body fat, sex and age as predictors, with ...


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Generally, the relationship between independent and dependent vars matters the most and this is what the author is doing here: Just as in univariable regression, the coefficient of determination describes the overall relationship between the independent variables Xi (weight, age, body-mass index) and the dependent variable Y (blood pressure). It ...


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The formula cited is simply the formula for the beta coefficient, or the standardized regression coefficient. Thus, the book is simply stating the formula for calculating standardized versions of c and d in the regression equation stated. Any stats program should be able to provide standardized regression coefficients. However, often unstandardized ...


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