Questions tagged [sums-of-squares]

sum of squares plays an important role in statistical models based on the normal distribution, like ANOVA.

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Fitting data to sums of squares of sinusoids

I am given some time series data $(t, y(t))$ sampled at regular intervals $t=0,s,2s,3s,\ldots,1$ for some step size $s$, obtained from a function $$ y(t) = y(t, \vec A, \vec \delta) = \sum_{i=1}^N ...
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Proof that more predictor variables will lead to less SSR [duplicate]

Say I have a multilinear model $y = X\beta - \varepsilon_1$, let's call this A. I also have another multilinear model $y = X_i\gamma - \varepsilon_2$, let's call this B. Where $X_i$ is a subset of $X$,...
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Which type of sums of squares does lm-function in R use?

I ran a two-way ANCOVA in R: ancova = lm(DV ~ IV1*IV2 + CV1 + CV2 + CV3, data = Data) summary.aov(ancova) Anybody know if this uses type III sums of squares? I ...
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How unbalanced is unbalanced for factorial ANOVA?

Colleagues and I conducted an experiment with participants randomly allocated to each condition using survey software. Unfortunately, the cell sizes ended up being more unbalanced than we anticipated: ...
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Is least squares means (lsmeans) statistical nonsense?

I recently came accross this quote from Brian Ripley, who seems to be well-regarded as a statistician. "Some of us feel that type III sum of squares and so-called ls-means are statistical ...
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Proof that Regression Sum of Squares and Residual Sum of Squares are independent random variables

Having consulted a number of sources, I still can't find a complete proof that Regression Sum of Squares ($SS_{regression}$) and ($SS_{residual}$) are independent random variables. I'll be doubly ...
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18 views

Shortcut for computing RSS at different split point when building regression trees?

I'm coding regression trees from scratch in R. For a given ordered predictor variable, $X$, obviously I have to compute the RSS at each unique ordered value of $X_i$. When moving over to the next ...
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Oneway ANOVA Sums of square calculation questions

From reading this explanation (https://www.r-bloggers.com/2011/03/anova-%E2%80%93-type-iiiiii-ss-explained/) of typeI-III anova sums of squares(SS) i'm under the impression that for a oneway/one ...
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Why contrast in ANOVA has one degree of freedom?

I thought I understood the degree of freedom after reading Wikipedia explanation, but came across the sum of squares for contrasts $\{c_i\}$ $$SS_C = \frac{(\sum_i {c_i \bar{y_i}})^2}{\sum_i c_i^2 /...
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Contrasts in ANOVA - need intuitions to understand its sum of squares and its degree of freedom

I have two questions about contrasts in ANOVA for testing the hypothesis that $\sum_i c_i y_i = 0$. How is the sum of squares for contrasts $\{c_i\}$, $SS_C = \frac{(\sum_i {c_i \bar{y_i}})^2}{\sum_i ...
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Distribution of the squared length of a multivariate normal vector [duplicate]

Let $\mathbf{X} \sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$ be a multivariate Normal variable with $d$ dimensions. I'm interested in the marginal distribution of $\|\mathbf{X}\|^2 = \sum_{i=1}^d ...
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What exactly is explained sum of squares, and why do we focus on minimizing error rather than maximizing explained variability?

In the third of my questions about decomposing the total sum of squares, I want to focus on the sum of squares of the regression. I can make sense of what the sum of squares of the residuals means: ...
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115 views

In a PCA setting, is there any relationship with the sum of squares of the scores (t1) with the eigenvalue of that principal component?

When one computes the vector of scores (t1) using the principal component (p1) the data is being projected over the direction of biggest variation. One could measure the distance between the point ...
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What is the lowest sum of squared errors (sse) of a dataset?

Given $X \subset R^d$ a set of data points, $z \in R^d$ a vector, $SSE(X)=\sum_{i=1}^{|X|}||X_i-z||^2$, I wonder what is the best value of $z$ so that $SSE$ is the lowest? I suspect that $z$ is the ...
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How are terms in SS (corrected sum of squares) independent?

What does the highlighted test mean? How are the terms not independent? Can someone elaborate in detail?
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Source for BIC with residual sum of squares?

Wikipedia states that under certain circumstances, $\mathrm{BIC} $ can be calculated as: $$\mathrm{BIC} = n \ln(\mathrm{RSS}/n) + k \ln(n),$$ where $\mathrm{RSS}$ is the residual sum of squares. ...
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Kruskal–Wallis one-way analysis of variance is related to what kind of regression?

One-way anova is similar to regular linear regression because both use the F-test which involves sums of squares among other reasons. Is Kruskal–Wallis one-way analysis of variance similar to some ...
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Difficulty with averaging corrected sample variances of different degrees of freedom:

I have a number of measurement samples of which some have 2 measurements and some have 3. I wish to make the most accurate estimation of population variance I can, and understand that ignoring data ...
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810 views

Residual sum of squares in a regression

I understand that in a linear regression model, the residual sum of squares will either remain same or fall with the addition of a new variable. What if the two models were $$ I \colon y_i=\...
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2answers
288 views

Simplifying the Matrix Form of the Solution to Ridge Regression

I'm trying to understand how to obtain the solution to an objective function by solving for the parameter vector $\theta$ in ridge regression. I found an example here from Naomi which takes an example ...
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What is the correct implementation of BIC with residual sum of squares?

BIC is most often calculated by maximizing the log likelihood function. However, it is also possible to calculate BIC with residual sums of squares. This is pretty easy to find online and not an issue ...
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Formula for type III sum of squares of the intercept term in linear multiple regression

assume we have the regression model: $$Y = b_0 + b_1 x_1 + \dots + b_k x_k + \varepsilon $$ I know the formulas for all type III sum of squares for the regression terms except the formula for SS of ...
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Coefficient of determination relationship?

$R^2 = \frac{SSREG}{SSTOT}$ or $R^2 = 1-\frac{SSRES}{SSTOT}$ If $X$ is the predictor random variable for science SAT and $Y$ is the predictor random variable science GPA given by equation $$\hat Y =...
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746 views

Sum of Squared Error Chi-Square distribution degree of freedom in Multilinear Regression

In this link it says that $Y$ variables has zero covariance (because covariance matrix has only diagonal terms) which implies they are independent. Actually in linear regression $Y$ takes its ...
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494 views

What is the difference between the residual, lack of fit and pure error In F test for Regression Analysis?

What is the difference between the residual, lack of fit and pure error and how to calculate each of them in ANOVA F test for nonlinear regression?
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Two way ANOVA, no difference between tests based on type I vs type II sums of squares

I'm learning about two way ANOVA possibly with interaction. I'm following this tutorial http://www.sthda.com/english/wiki/two-way-anova-test-in-r This is their code. ...
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Calculating SSE in R

I'm following the "Intro to Statistics" course in Data Camp and I'm having some trouble as it seems that the course is contradicting itself: https://s3.amazonaws.com/assets.datacamp.com/production/...
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Why is regression line represented as $y = b0 + b1 * x$?

I am new to Data Science and ran into Regression Line formula which is $ y = b0 + b1 *x $ (where x is dependent variable, y is predicted variable) I understood the meaning of this formula as a ...
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394 views

Better to Minimize Absolute Error or Sum of Squared Error?

I have an Excel model which predicts the number of customers for a given month. The prediction depends on a churn rate. I have the absolute error (actual vs predicted), along with squared error and ...
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Why error sum of squares has n-2 df (possibly not duplicate, please read on)? (Regression Question Series - Part 4)

In simple linear regression, the error sum of squares is given by $$ \text{SSE} = \sum_{i=1}^n(y_i - \hat{y_i})^2 \\ \hat{\sigma}^2 = s^2 = \dfrac{\text{SSE}}{n-2} $$ where $n-2$ is the degrees of ...
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170 views

Minimize Logged Sum of Squares?

When numerically maximizing the likelihood function it is standard practice to do this indirectly by minimizing the negative log-likelihood. When numerically minimizing the residual sum of squares (...
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R: anova(lm): What is the Sum Sq when we have two inputs

What is the formula to calculate the Sum sq column for the inputs? Answers to some other questions state, that it should be $RSS = \sum (\hat Y_i -\bar Y)^2$, yet this is false, it only gives the ...
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2answers
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Showing that $\sum_{i=1}^n (y_i-\hat{y_i})(\hat{y_i} - \bar{y}) = 0$ for the generalized linear model [closed]

Exercise : Prove that for the generalized linear model, it is : $$\sum_{i=1}^n (y_i-\hat{y_i})(\hat{y_i} - \bar{y}) = 0$$ Question : How would one proceed with proving that for the generalized ...
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Squaring floats between -1 and 1 reduces sum of squares, so why do it? [duplicate]

I have been learning basic statistical testing as it relates to agriculture and have become familiar with the common practice of summing squared raw deviation values, whether in something simple like ...
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2answers
216 views

Calculating F-statistic, why is SS used instead of just $r^2$?

The calculations and question are for a simple regression (one independent and one dependent variable). SSres = SSy * (1 - $r^2$) SSreg = SSy * $r^2$ F = (SSreg / DFreg)/(SSres / DFres) So, I've ...
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420 views

Understanding ANOVA as regression / type III SS in R

I'm confused about how type III SS are calculated for a "main effect". According to what I have read, Type III SS is calculated by evaluating the change in the SSE by removing only the variable in ...
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1k views

2-way ANOVA in Python statsmodels yields different Sum of Squares than SPSS

I am learning to use Python for my statistical analyses, and while figuring out how to perform a 2-way ANOVA with statsmodels I found that my Python code yielded slightly aberrant values. Comparing ...
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In regression, when partitioning SS among predictors, what determines which predictors get the SS that can be attributed to more than one predictor?

In regression analysis, predictors sometimes correlate (and in my field, psychology, they always do; often because they are simply measurements of the same aspects of human psychology). If predictors ...
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How should Type II SS be calculated in a mixed model?

I have a data set (and corresponding mixed model) which gets very different p-values for one of the two-way interactions when tested using Type I (sequential, taking care that it's last), and Type II (...
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Deriving the F-test from ${{SSE_R-SSE_F}\over{(n-q)-(n-p)}}/{{SSE_F}\over{n-p}}$

Given a Full and Reduced model, the F-test to see if the reduced model is significant is given by $$ {{SSE_R-SSE_F}\over{(n-q)-(n-p)}}/{{SSE_F}\over{n-p}} $$ I'm trying to understand how this is ...
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Find the Sum of Squares and using them complete an F-test for goodness of fit

I have this question ( would be grateful if someone could put it on for me) https://gyazo.com/5afe5d7c9d12acdff8f9c55db74f97c9 And I am concerned with part b). I know what $RSS = SSE$ is due to the ...
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Which one to choose Type-I, Type-II, or Type-III ANOVA? [duplicate]

I don't understand what the difference is between TypeI and TypeIII? Since my background is not very mathematics, it's very difficult for me to understand this mathematical notation; Type I SS: SS(A) ...
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206 views

Does forward model selection by $R^2$ or SSR differ?

I'm trying to implement forward selection and need to add a feature only if it will make the sum of squared residuals (SSR) lower. I am using Python's statsmodels ...
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The Sums of squares in 3 way Anova in R output has changed as the order of the variables in the modes changes [duplicate]

I am trying to fit a 3 way ANOVA using R. Then i realized the sums of squares of the output has changed when the order of the variables in the model are changed.Can anyone figure out the reason for ...
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331 views

Mixed-effect model single term deletion — should I change my random effects?

In short I recently had a little conversation on the lme4 project's GitHub on how to properly test the significance of effects in a mixed-effect model, which made ...
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242 views

Test significance using Type II ANOVA following Type III ANOVA?

I hope you can help me with a theoretical question about how to proceed in my analysis. I have come across many posts discussing differences between Type II and III SS Anovas, however, because I don't ...
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Trying to understand hierarchical testing of nested regression models

I've learned multiple regression, but never did I learn hierarchical regression (i.e., the hierarchical testing of nested regression models) before. Based on the image attached, I assume because the ...
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1k views

In regression analysis, How can R^2 represent the total explained variance, if it can be computed from an equation with only unique contributions?

Background In regression analysis, $R^2$, the squared multiple correlation, represents the proportion of explained variance by the regression model. Most software's default setting uses Type-III sums ...
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Is Square Root of the Variance of a Regression Coefficient the Standard Error?

Quick question, in the textbook "Introductory Econometrics", the variance of a Regression Coefficient is given as: $var(\hat\beta_j) = \frac{\sigma^2}{SST_j(1-R_j^2)}$ where, $SST_j$= $\sum_{i=1}^n (...
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2answers
242 views

Why do type III sums of squares require orthogonal contrasts?

I have read many times that one has to set orthogonal contrast to get correct type III sums of square. E.g. John Fox says To compute Type-III tests using incremental F-tests, one needs contrasts ...