# 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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### ANCOVA $\eta^2$ definition

I am struggling to find references to understand what is the exact definition of $\eta^{2}$ is two way anova.
Let's say I am estimating the following ANCOVA model:
$$
y_{ijk} = \mu + \alpha_i + \...

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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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### 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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### 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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### 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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### 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: http://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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195 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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### 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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### 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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### 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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158 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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697 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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### What's the difference between the SS in the variance and the TSS?

I'm trying to understand how these two statistics differ.
My understanding is the variance is the sum of squares of the predictor divided by the degrees of freedom.
On the other hand, the sum of ...

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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)
http://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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### 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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### 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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### 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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### 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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### 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 ...

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### Obtaining Information criterion or log likelihood from a model's squared error

I am concerned with the formulas I've seen for the AIC or BIC when using a squared-errors instead of likelihood.
On the AIC wikipedia page, there is the cryptic formula for LL:
$ -{\frac {n}{2}}\ln(...

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379 views

### Is equation for total sum of squares always valid?

I wanted to do simple calculations for sum of squares for linear regression using a very simple example.
1) Let's assume that we have 3 obervations [x,y]: [[1,1], [2,2], [3,3]].
2) I created simple ...

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### How to calculate the mean and standard deviation of weighed Noncentral chi distribution?

I would like to calculate the mean and standard deviation of :
Dist=(X^2+Y^2+Z^2)^0.5
Knowing that X, Y and Z are normal noncentral and nonstandard variables. they have a mean different from 0 and ...

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### Motivation for Ward's definition of error sum of squares (ESS)

Ward (1963) provides a commonly used criterion for hierarchical clustering. It's based on the following definition (p. 237):
Given a set of ratings for 10 individuals, $\{2, 6, 5, 6, 2, 2, 2, 0, 0, ...

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### Basics relative to chi square, likelihood, fits,

I'm confused to separate all the different meanings and connections.
The background of my question: On the one hand related to lmer models and on the other hand to the goodness of a fit. And their ...

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161 views

### How is SSerror calculated in a factorial design?

I have found that
SSerror = SStotal - SStreatment
which makes sense to me alongside an example of a one-way ANOVA, however, how does this apply to a factorial ANOVA?

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### Sum of Squares in Randomized Complete Block design

I'm interested in the Randomized Complete Block design (RCB) with
three treatments, which I call $A$, $B$, and $C$,
and four blocks ($b=4$):
\begin{align}
\textrm{block 1: } (y_{A,1}; y_{C,1}; y_{B,...

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### What is the physical intuition behind the equality $\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar x) x_i$?

Suppose that $x_1,\ldots,x_n$ are real numbers, and let $\bar x$ denote the average $\frac{\sum_i x_i}n.$ I know how to prove on paper that the equality
$$\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar ...

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### ANOVA Type II vs Type II SSq for multiple factors

I have pondered type II vs type III sums of squares for some time and looked at posted examples of calculations for 2 main factors (e.g. this very nice page). That is, in the case of a model:
X = A + ...

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### What is the total sum of squares for these points? [closed]

y = (-2,2,3,4) and x = (3,5,8,12). Lin. Model is y = -2.5 + 0.6x
Please explain how you got the answer. Thank you!
I am confused on whether I should be calculating based on the x or y values, or ...

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### Generalized eta squared

Partial eta-squared are very often used in psychological litterature. As underlined by some authors (e.g., Baguley, 2009; Bakeman, 2005; Olejnik & Algina, 2003), this standardized measure of ...

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363 views

### Distribution of $(n-2)MSres/\sigma^2$ in simple linear regression

In simple linear regression, it was proved that $(n-2)MS_{RES}/\sigma^2$ follows a $\chi^2_{n-2}$ distribution.In order to prove this,the fact that $e_i$ or $y_i - \hat{y_i}$ follows a Normal ...

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### Feature scaling getting high Error Value (sum of squared)

I got a test data set to work on in order to implement feature scaling and get better results with gradient descent.
It looks like this:
...

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### Derivation of SSE(reduced)-SSE(full) = $(C\hat \beta - h)'(C(X'X)^{-1}C)^{-1}(C\hat \beta - h) $

When stating hypothesis in matrix formulation, as H0: $C\hat\beta=h$, $SSE(reduced)-SSE(full)$ can be expressed as:
$$(C\hat \beta - h)'(C(X'X)^{-1}C)^{-1}(C\hat \beta - h) $$
How is this result ...

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### Residual sum of squares, two different equations, why are they equal?

One definition of the Residual Sum of Squares is:
$$
S_r = (y-X\hat{\beta})^T(y-X\hat{\beta})
$$
And I think I understand it.
Now I have seen a different definition:
$$
S_r = y^Ty- \hat{\beta}^TX^...