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The model being $$y=a+bx+\frac c {\sqrt x +d}$$ it is nonlinear with respect to the parameters (because of $d$) and adjusting coefficients $a,b,c,d$ will require nonlinear regression and this will also require some reasonables estimates of the parameters for starting it. What you could notice is that, for a fixed value of $d$, the model is linear. For a ...


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$$\sum_{i=1}^n x_i^2=x_1^2+x_2^2+\cdots+x_n^2$$ $$\Big(\sum_{i=1}^n x_i\Big)^2=(x_1+x_2+\cdots+x_n)^2$$


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$$\left(\sum x_i\right)^2 = \sum x_i^2 + 2\sum_{i<j}x_i x_j$$


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You may observe that $$ \hat{\beta}_1 =\frac{\sum_{i=1}^n x_i y_i - n \bar{x}\bar{y}}{\sum_{i=1}^n x_i^2 -n\bar{x}^2} =\frac{\frac1n\sum_{i=1}^n x_i y_i - \bar{x}\bar{y}}{\frac1n\sum_{i=1}^n x_i^2 -\bar{x}^2}.\tag1 $$ Then you may prove that $$ \frac1n\sum_{i=1}^n \left(x_i -\bar{x}\right)^2=\frac1n\sum_{i=1}^n x_i^2 -\bar{x}^2 \tag2 $$ and that $$ ...


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For example, let $x_i$ represent $i^{th}$ natural number, and n = 4. Then $\left(\sum_{i=1}^n x_i\right)^2$ = (1 + 2 + 3 + 4)^2 = (10)^2 = 100 and $\left(\sum_{i=1}^n x_i^2\right)$ = (1^2 + 2^2 + 3^2 + 4^2) = (1 + 4 + 9 + 16) = 30.


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For a coxph object (the Cox regression) you can use the extractAIC command from the stats package. For more details see here.


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You are looking for a technique called "Least-squares regression". This method allows for overdetermined systems (more equations than variables) to be fit to the polynomial that minimizes the sum of the distances (actually the sum of the squared distances) from each point to that polynomial. You specifically want the matrix version that looks like this ...


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Let us suppose that you have $n$ data points $(A_i,B_i,I_i)$ ($i=1,\cdots,n)$ and you want to find the "best" $x$ and $y$ for matching the model $$I=x A+y B$$ the genral method, as already answered by G-Cam, is ordinary least square which consist in the minimization of $$F=\sum_{i=1}^n \big(x A_i+y B_i-I_i\big)^2$$ Computing the derivatives and setting them ...


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The studentized residuals are $$t_i=\frac{\epsilon_i}{\hat\sigma\sqrt{1-h_{ii}}}$$ Where $\epsilon_i$ is the residual, $h_{ii}$ the leverage and $\hat\sigma$ is the estimate of the standard deviation of residuals, that is $$\hat\sigma^2=\frac1{n-m}\sum_{i=1}^n\epsilon_i^2$$ Where $n$ is the number of observations (here $4$) and $m$ the number of ...


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It does not seem to be correct. Let $A,B,C$ be random variables such that $P(A=1)=P(A=0)=P(A=-1)=1/3$, $C=A$, and $B=A^2$. Observe that $E[A]=E[C]=0$ and $E[B]=2/3$. Then, letting $\rho$ denote the covariance, $$ \rho_{A,C} = Var(A)= \frac{1}{3}(1-0)^2+\frac{1}{3}(0-0)^2+\frac{1}{3}(1-0)^2=\frac{2}{3} $$ On the other hand, $$ \rho_{A,B} = \rho_{B,C} = ...


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$Cov(X,Y) = \sigma_X\sigma_Y\rho_{X,Y}.$ Where $\rho$ is the correlation. Usually, you see this written the other way round to define correlation. $\beta_{X/Y} = \frac{\sigma_X}{\sigma_X}\rho_{X,Y}$ $\beta_{A/B}\beta_{B/C} = \frac{\sigma_A}{\sigma_C}\rho_{A,B}\rho_{B,C}$ $\beta_{A/B}\beta_{B/C} = \beta_{A/C}$ would imply that $\rho_{A,B}\rho_{B,C} = ...


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Let $\eta_j(i)$ denote the $i$th element of vector $\eta_j$ with $j =1,2$. By definition, we have $\text{cov}(\eta_1,\eta_2) = \left( \begin{array}{cc} cov(\eta_1(1),\eta_2(1)) & cov(\eta_1(1),\eta_2(2)) \\ cov(\eta_1(2),\eta_2(1)) & cov(\eta_1(2),\eta_2(2)) \end{array} \right),$ where direct substitution gives $\text{cov}(\eta_1,\eta_2) = ...


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Yes, it is meaningful to imagine the regressors are sampled from a theoretical distribution. The quantity $P(X_1=0)$ would then refer to the theoretical distribution for $X_1$, not to the observed values $x_{1,1},\ldots,x_{1,p}$ of $X_1$. Note that in elementary treatments of regression we assume the regressors are non-random, i.e., they are known constants ...


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Your interpretation of a prediction interval is incorrect. A 90% prediction interval will contain 90% of the probability of the true underlying distribution on average (not always nor "at a minimum"). What you are thinking about is a less-taught concept: a Tolerance Interval. A tolerance interval is specified by a confidence and a coverage: The coverage is ...


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A few more ideas, in addition to the ones you mentioned: Direction. Fit a straight line to the curve, and then look at the direction of the line. Direction can be expressed by two angles. Moments of inertia. These measure the "distribution" of matter in the curves, in some vague sense. Some things with bounding boxes. For example, the ratio of the ...



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