# 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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### cannot reproduce the example for nested anova

I'm working on the example for nested ANOVA from the Handbook of Biological Statistics. Copy-pasteable data can be found at https://www.biostathandbook.com/nestedanova.html and below shows my ...
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### What exactly do Anova sums of squares represent in lmer models?

In a lme4::lmer model, the anova function returns an sequential analysis of variance table. As explained on page 34 of http://...
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### Interpretation of default Type III Sums in Squares in R

Many posts (i.e. here and here) discuss how the Type III sums of squares produced by car::Anova() in R are incorrect or nonsensical under R's default model parameterization, "contr.treatment"...
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### How is the matrix of corrected sum of squares and products defined?

In the article Johns, S. "On identifying the population of origin of each observation in a mixture of observations from two normal populations." Technometrics 12.3 (1970): 553-563, the ...
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### Summation of terms in survival function giving negative probability

The following expression represents a survival function for a group size (n), where survival is a percentage on a scale of 0 to 100. $$p(x) = 100 - \beta(x \cdot x) = 100 - \beta\sum_{i=1}^{n}x_i^2$$...
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### Does Residual Sum of Squares always decreases with an additional independent variable? [duplicate]

^ What the question says. Trying to determine whether this statement is true or false. 8 views

### Is there any formal proof that in R type-1 anova() run twice on 2 models with switched terms equals type-2 (looking at the last entry)?

I observed something interesting to me. Let's assume that I have an unbalanced, additive model (no interactions) with 2 terms. When I swap the order of the two terms and run type-1 ANOVA over them, ...
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### Synonyms of Residual Sum of Squares

As a novice, I'm finding it difficult to learn statistics, partly because there are often many different words for the same thing. When I'm reading about stats, I don't realize that the thing I'm ...
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### "Normalized mean squared error" says WHAT?

I know that mean squared error is a public and popular metric to evaluate the efficiency of the model and architecture. Also, it is the tool to evaluate the result in such if, the ...
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### Why type III sum of squares is not recommended to use when there is significant interaction?

I did notice that in the following example, I obtained a insignificant main effect for treatment when including interaction, although the treatment is clearly effective. Any intuitive explanation? ...
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### Why calculate Mean Squared Error for Regression?

Once we fit the model to the points why is it necessary to find the mean squared error? What happens if I don't calculate Mean Squared Error?
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### Which type of sum of squares corresponds to the partition of variance formula for two-way ANOVA?

Does anyone have a good reference for the formula for type I and type III sum of squares in two-way ANOVA? I learned the following formula for partition of total sum of squares in two-way ANOVA. Is it ...
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### The number of clusters in the K-means and the within-cluster SS

Given a collection of observatons $\{X_i\}_1^N$ and prespecify the number of clusters K. The K-means solves $$\underset{\{C_k\}_1^K}{\arg\min} \sum_{k=1}^K \sum_{i \in C_k}|| X_i - \mu_k||^2$$ where ...
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### Conceptual/Math/brainteaser Question: Multiple Linear Regression

This is kind of a brainteaser and I'm struggling to solve it, any ideas on approaching this would be valued: I thought about using substitution of the different x and z into the third regression ...
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### Confused: Why is lme4 changing techniques from Wald F tests to Wald Chisquare?

I've constructed some LMEMs that use dichotomous variables and their interactions as regressors, and I've become confused by the output. When I only assess a single interaction, using the code below (...
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### 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 ... 73 views

### 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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### 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: 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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### 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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### 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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