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You cannot identify confounders in this way. This is because the same pattern of results could appear if $Z$ was a confounder or not. If $Z$ was a mediator or a collider, for example, you would see the coefficient on $X$ change between the two models. Also, if there is confounding by other unmeasured variables, the coefficient on $X$ may change regardless of ...


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With some slightly different notation you can make the problem easier to interpret. It seems like you are looking to maximize the correlation between $$\begin{array}{rclcrcl} P_j &=& \sum_{i=1}^n P_{ij} V_i & \quad \text{and} \quad & T_j &=& \sum_{i=1}^m T_{ij} W_i \end{array}$$ The solution to maximize the correlation is the same as ...


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Multiply the regression coefficient and the confidence limits on the log-odds scale by $a$ and exponentiate to get the odds ratios and the corresponding confidence interval for an increase of the predictor by $a$. The $p$-value is not affected by scaling. Alternatively, dividing the predictor by $a$ before fitting the model will scale the corresponding ...


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Why doesn't the effect of old variables remain the same when including a new variable? Reading your question again, I believe that you are referring to the previous point "the effect of X on Y when other independent variables were hold constant" since you mentioned earlier that you understood the change in coefficients. So, what is described here ...


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Can I use the mean of the estimates of coefficients for forecasting out of sample? You could, but why? A big selling point of Bayesian modelling is the ability to integrate over uncertainty in the parameters. You would do this by generating from the posterior predictive distribution, namely $$ p(\tilde{y} \vert y) = \int p(\tilde{y}\vert \theta) p(\theta \...


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Basically, when you add variables to a regression, no matter what those variables are, the coefficient on exposure can increase or decrease. You generally cannot learn anything about the causal structure of your system by observing coefficient change when adding or removing covariates to a model. Often, adding a mediator will decrease the coefficient on the ...


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Testing whether the effect of condition A is different from the effect of condition B is the same as testing whether the mean of condition A is different from the mean of condition B. You can do this in a standard one-way ANCOVA (adjusting for pre-treatment scores), where you specify a contrast between A and B or just perform all pairwise comparisons.


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Yes, you can use the $t$-statistic to compare relative significance. The $t$-statistic is scale-independent, so the absolute value of large vs. small values, i.e., $|t_j|$, across coefficients will reflect relative significance. For regression, the t-statistic for the $j$th predictor variable is \begin{equation} t_j=\frac{\beta_j}{\mathrm{s.e.}(\beta_j)}, \...


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You can't. Logistic regression does not have an analytical or a closed form solution When is logistic regression solved in closed form?


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