Search

Extra Sum of Squares
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Multivariable_Regression
When there are many predictors, it is often of interest to see if one or a few of the predictors can do the job of estimation of the mean response and prediction of new observations well enough. This ...When there are many predictors, it is often of interest to see if one or a few of the predictors can do the job of estimation of the mean response and prediction of new observations well enough. This can be put in the framework of comparison between a reduced regression model involving a subset of the variables versus the full regression model involving all the variables.
Test for Lack of Fit
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Simple_linear_regression/Test_for_Lack_of_Fit
The null hypothesis in which the linear model holds is: $H_{0}: \mu_{j} = \beta_{0} + \beta_{1}X_{j}$ , for all $j = 1, ..., c$ . Let $\bar{Y} = \frac{1}{n_{j}} \sum_{i = 1}^{n_{j}} Y_{ij}$ , and \...The null hypothesis in which the linear model holds is: $H_{0}: \mu_{j} = \beta_{0} + \beta_{1}X_{j}$ , for all $j = 1, ..., c$ . Let $\bar{Y} = \frac{1}{n_{j}} \sum_{i = 1}^{n_{j}} Y_{ij}$ , and $\bar{Y} = \frac{1}{c}\sum_{j=1}^{c}n_{j}\bar{Y}_{j} = \frac{1}{n}\sum_{j=1}^{n}\sum_{i=1}^{n_j}Y_{ij}$ , where $n = \sum_{j=1}^{c}n_{j}$ . $SSPE = SSE_{full} = \sum_{j=1}^{c}\sum_{i=1}^{n_j}(Y_{ij} - \bar{Y}_{j})^2 = \sum_{j=1}^{c}\sum_{i=1}^{n_j}Y_{ij}^2 - \sum_{j=1}^{c}n_{j}\bar{Y}_{j}^2$
Analysis of variance approach to regression
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Simple_linear_regression/Analysis_of_variance_approach_to_regression
Under $H_0, F^*$ has the $F$ distribution with paired degrees of freedom (d.f.( SSR ), d.f.( SSE )) = (1, n - 2 ), (written $F^* \sim F_{1, n - 2}).$ Thus, the test rejects $H_0$ at level of...Under $H_0, F^*$ has the $F$ distribution with paired degrees of freedom (d.f.( SSR ), d.f.( SSE )) = (1, n - 2 ), (written $F^* \sim F_{1, n - 2}).$ Thus, the test rejects $H_0$ at level of significance $\alpha$ if $F^* > F( 1 - \alpha; 1, n - 2 ),$ where $F( 1 - \alpha; 1, n - 2 )$ is the $(1 - \alpha )$ quantile of $F_{1; n - 2}$ distribution.
Least squares principle
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Simple_linear_regression/Least_squares_principle
The quantity $f_i(\widehat\beta)$ is then referred to as the fitted value of $Y_i$ , and the difference $Y_i - f_i(\widehat\beta)$ is referred to as the corresponding residual. If the functions \...The quantity $f_i(\widehat\beta)$ is then referred to as the fitted value of $Y_i$ , and the difference $Y_i - f_i(\widehat\beta)$ is referred to as the corresponding residual. If the functions $f_i(\beta)$ are linear functions of $\beta$ , as is the case in a linear regression problem, then one can obtain the estimate $\widehat\beta$ in a closed form.
Analysis of Variance
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Analysis_of_Variance
where $SSTO = \sum_{i=1}^{r}\sum_{j=1}^{n_i} (y_{ij} - \overline{y}_{..})^2$ is the Total Sum of Squares; $SSE = \sum_{i=1}^{r}\sum_{j=1}^{n_i} (y_{ij} - \overline{y}_{i.})^2$ is the Error Sum of ...where $SSTO = \sum_{i=1}^{r}\sum_{j=1}^{n_i} (y_{ij} - \overline{y}_{..})^2$ is the Total Sum of Squares; $SSE = \sum_{i=1}^{r}\sum_{j=1}^{n_i} (y_{ij} - \overline{y}_{i.})^2$ is the Error Sum of Squares and $SSTR = \sum_{i=1}^{r}n_i(\overline{y}_{i.} - \overline{y}_{..})^2$ is the Treatment Sum of Squares.
Experimental Design
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Experimental_Design
Experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not.
Analysis of Factor Level Means and Contrasts
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Experimental_Design/Analysis_of_Variance/Analysis_of_Factor_Level_Means_and_Contrasts
A 100(1- $\alpha$ %) two sided confidence interval of $\mu_{i}$ is given by $\bar{Y}_{i\cdot} \pm s(\bar{Y}_{i\cdot})t(1 - \frac{\alpha}{2}; n_{T} - r)$ where \(t(1 - \frac{\alpha}{2}; n_{T} - r)\...A 100(1- $\alpha$ %) two sided confidence interval of $\mu_{i}$ is given by $\bar{Y}_{i\cdot} \pm s(\bar{Y}_{i\cdot})t(1 - \frac{\alpha}{2}; n_{T} - r)$ where $t(1 - \frac{\alpha}{2}; n_{T} - r)$ denotes the $1 - \alpha/2$ quantile of the t-distribution with $n_{T} - r$ degrees of freedom. A contrast is a linear combination of the factor level means: $L = \sum_{i = 1}^{r}c_{i}\mu_{i}$ where $c_{i}$ 's are prespecified constants with the constraint: $\sum_{i=1}^{r}c_{i} = 0$ .
Simple Linear Regression (with one predictor)
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Simple_linear_regression/Simple_Linear_Regression_(with_one_predictor)
This gives us the normal equations: $nb_0 + b_1\sum_{i=1}^nX_i = \sum_{i=1}^nY_i$ $b_0\sum_{i=1}^nX_i+b_1\sum_{i=1}^nX_i^2 = \sum_{i=1}^nX_iY_i$ Solving these equations, we have: \[b_1=\frac{\sum_...This gives us the normal equations: $nb_0 + b_1\sum_{i=1}^nX_i = \sum_{i=1}^nY_i$ $b_0\sum_{i=1}^nX_i+b_1\sum_{i=1}^nX_i^2 = \sum_{i=1}^nX_iY_i$ Solving these equations, we have: $b_1=\frac{\sum_{i=1}^nX_iY_i-n\overline{XY}}{\sum_{i=1}^nX_i^2-n\overline{X}^2} = \frac{\sum_{i=1}^n(X_i-\overline{X})(Y_i-\overline{y})}{\sum_{i=1}^n(X_i-\overline{X})^2}, b_0 = \overline{Y}-b_1\overline{X}$
Regression diagnostics for one predictor
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Simple_linear_regression/Regression_diagnostics_for_one_predictor
Hence, denoting by $b_1^{(n-1)}$ the least squares estimate of $\beta_1$ computed from the first n-1 observations, we have $$b_1^{(n-1)} = \frac{\sum_{i=1}^{n-1} (X_i - \overline{X}_{n-1})(Y_i - \...Hence, denoting by $b_1^{(n-1)}$ the least squares estimate of $\beta_1$ computed from the first n-1 observations, we have $b_1^{(n-1)} = \frac{\sum_{i=1}^{n-1} (X_i - \overline{X}_{n-1})(Y_i - \overline{Y}_{n-1})}{\sum_{i=1}^{n-1}(X_i - \overline{X}_{n-1})^2} = \frac{420}{460} = 0.913.$ For the whole data set, $\overline{X} = (\sum_{i=1}^{n-1} X_i + X_n)/n = 7. \overline{Y} = (\sum_{i=1}^{n-1} Y_i + Y_n)/n = 22.$
Simple linear regression
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Simple_linear_regression
The basic problem in regression analysis is to understand the relationship between a response variable, denoted by $Y$ , and one or more predictor variables, denoted by $X$ . The goal is to describe...The basic problem in regression analysis is to understand the relationship between a response variable, denoted by $Y$ , and one or more predictor variables, denoted by $X$ . The goal is to describe this relationship in the form of a functional dependence of the mean value of Y given any value of $X$ from paired observations $\{(X_i,Y_i) : i=1,\ldots,n\}$
Multiple Comparison
https://stats.libretexts.org/Bookshelves/Computing_and_Modeling/Supplemental_Modules_(Computing_and_Modeling)/Regression_Analysis/Analysis_of_Variance/Multiple_Comparison

Search

Text Color

Text Size

Margin Size

Font Type

Support Center

How can we help?