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Partition of sums of squares (ANOVA)

Can anyone explain the theory (or the formula) about computing Sum Sq (bold highligh below) related to regression items? The Wikipedia link gives an introduction on how to calculate the total, model, ...
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0answers
44 views

How can I make 2-way ANOVA (Type I, II and III) with R?

I have the following data with factor 1 (A, B and C) and factor 2 (D and E): $$ \begin{array}{ccc} \hline & D & E\\ \hline A & 68 & 59\\ & 65 & 57\\ & 63 & 54\\ ...
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1answer
22 views

On the connection between SSE and absolute deviation from the centroids

Is there any connection between sum of squared error SSE and the absolute deviation from the centroids after clustering. More formally, I have clustered $T=\{x_i\}, i\in\{1,\ldots,n\}$ and the ...
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1answer
33 views

Squared sum of errors

Why does $\mbox{SST} = \mbox{SSR} + \mbox{SSE}$ hold? It is understandable that $\sqrt{\mbox{SST}} = \sqrt{\mbox{SSR}} + \sqrt{\mbox{SSE}}$ but how can $\mbox{SST} = \mbox{SSR} + \mbox{SSE}$ hold? ...
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1answer
54 views

First order condition of sum of squares with respect to variance of residuals

Consider the criterion function for ordinary least squares $$ S(b)=(Y-X'b)'(Y-X'b) $$ with Y, a matrix of dependent variables, and X, a matrix of explanatory variables. It is of course known that: ...
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1answer
59 views

Decomposing total sum of squares

Consider the general linear regression model: $$y_i = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + \cdots + \beta_px_{ip} + \epsilon_i = \mathbf{x}_i^t \beta + \epsilon_i$$ where $\textbf{x}_i = ...
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1answer
97 views

Least Squares Fits to Experimental Data

My attempt at making sense of the problem: The problem provides us with the sum of squares (SS; I believe that's the $18.1$). We can use the SS along with the number of samples $(21)$ to get the ...
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1answer
621 views
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1answer
37 views

Regarding sum of squares two ways how are they connected

$$ \sum(X^2) - \frac{(\sum X)^2}{n} = \sum(X^2) - m\sum X $$ This was nicely derived in Sum of squares two ways, how are they connected? But why is the second term called the "correction term for the ...
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1answer
103 views

Calculating sum of squares between groups

I have carried out this ANOVA: ...
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0answers
59 views

Degrees of freedom in regression analysis

I'm studying regression analysis but I'm struggling with really understanding how degrees of freedom are calculated. For example, if we have the simple scenario where $Y_i=\beta_0+\beta_1 X_i + ...
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0answers
25 views

Reading and interpreting the scatter matrix [duplicate]

The scatter matrix is defined as $$S = \sum_{j=1}^n (\mathbf{x}_j-\overline{\mathbf{x}})(\mathbf{x}_j-\overline{\mathbf{x}})^T$$ The trace (sum of the diagonal elements) of this matrix is ...
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1answer
40 views

Sum of squares proof where $N=n_{UC}+n_{TX}$

$TX$ is variable that indicates treatment status ($TX=1$ if the patient gets the new treatment, and $0$ otherwise, and $UC = 1 - TX$ indicates they got the standard treatment). Of $N$ patients, ...
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1answer
176 views

SSB - Sum of squares between clusters

I got a little confused with the squares and the sums. As far as I know, the variance or total sum of squares (TSS) is smth like $\sum_{i}^{n} (x_i - \bar x)^2$ and the sum of squares within (SSW) ...
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2answers
189 views

Sum of squared difference and Gaussian noise model

I have been reading that when the underlying error is distributed normally, then minimising the sum of squared difference between the observed data and the model is the appropriate cost function to do ...
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2answers
1k views

Distribution of sum of squares of T-distributed random variables

I am looking at the distribution of the sum of squares of T-distributed random variables, with tail exponent $\alpha$. Where X is the r.v., the Fourier transform for $X^2$, $\mathscr{F}(t)$ gives me a ...
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0answers
51 views

Independence between residual sum of squares and mean?

Let $Y_1, \cdots, Y_n$ be independent random variables each with the distribution $N(\mu, \sigma^2)$, then \begin{align*} S^2 &= \sum(Y_i - \overline{Y})^2 \\ &= \sum((Y_i - \mu) - ...
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0answers
99 views

Explain the difference between type 1,2,3,4 sums of squares

Can someone explain the difference between type 1:4 sums of squares using examples with relatable variables?
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0answers
250 views

Total Sum of Squares (TSS) - Exponential Regression

The total sum of squares is expressed as below [1][2], \begin{align} \rm{TSS} &= MSS + ESS \end{align} \begin{align} \sum_{i=1}^n (y_{i}-\overline{y})^2 &= \sum_{i=1}^n ...
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1answer
189 views

Does the standard error in OLS not need to be corrected by n?

The standard error of an estimator is defined as the square root of the the estimator variance (or mean squared error, MSE, for unbiased estimators). More specifically, if we wanted to get the ...
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0answers
118 views

Sum of residuals squared

How can I write the sum of squared residuals as a function of the sample mean and variance of $y$, given that the regression equation is: $y = \beta_0 + \beta_1(x-\bar{x}) + \epsilon$ where ...
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0answers
70 views

Can I use pool.scalar in mice (R) to pool test statistics?

After obtaining multiply imputed datasets and performing the desired analysis (in my case a repeated measures ANOVA), I want to pool test statistics (such as sum of squares or F-statistics) in order ...
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1answer
2k views

Why do we use a one-tailed test [F-test] in analysis of variance (ANOVA)?

Can you give the reason for using a one tailed test in the analysis of variance test? Why do we use a one-tail test - the F-test - in ANOVA?
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1answer
88 views

Partitioned sum of squares

I'm trying to calculate partitioned sum of squares in a linear regression. In the first model, there are two predictors. In the second model, one of these predictors in removed. In the model with two ...
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1answer
181 views

ANOVA sum of squares between groups

I'm trying to get an intuitive understanding of why the sum of squares between groups needs to be multiplied by the number of observations within each group. Using the ...
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1answer
65 views

Looking for simple examples of how to calculate type III and type IV SS

I have data collected on five species of fish at half a dozen locations in a lake over four years. The categories are not (at all) fully crossed, and I have a lot of empty cells due to logistical ...
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3answers
2k views

Choice between Type-I, Type-II, or Type-III ANOVA

We have a dataset with three variables (dV: self-reported measure on scale 1-5, assumed to be metric; iV1: factor with 4 levels; iV2: factor with 8 levels). We are interested whether the dV differs in ...
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0answers
75 views

Are the sequential sum of squares appropriate when treatments must be applied in sequence?

I'm working with some modeled future stream flow data that was created in two steps. First, future precipitation predictions (at certain points in a watershed) were created by running historical ...
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2answers
711 views

Why is the k-means algorithm minimizing the within cluster variance?

I have read that the k-means algorithm tries to minimize the within cluster sum of squares (or variance). With some brainstorming, a question popped up. Why is it that k-means or any other clustering ...
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0answers
73 views

How to Partition An Interaction SS in ANOVA

I'm a new user of R and I'm trying to replicate Table 6.18 on page 262 in Statistical Procedures in Agricultural Research, By K. A. Gomez and A. A. Gomez. New York, Chichester, etc.: Wiley (1984). ...
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1answer
330 views

Type III sums of squares

I have a linear regression model with one categorical variable $A$ (male & female) and one continuous variable $B$. I set up contrasts codes in R with ...
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1answer
1k views

R, Differences between lm and aov

What does explain the differences in p.values in the following aov and lm ? Is the difference only due to different types of Sum of square calculations ? ...
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1answer
146 views

Assessing quality of clusters

I am running into problem of assessing the quality of clusters. In my case, I have plotted the data after determining data classes. For each class (1, 2, 3) there are two clouds that appear separately ...
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1answer
107 views

Monette & Fox's conditional hypotheses

G. Monette and J. Fox provide in these slides a framework for the Type II Analysis of Variance/Deviance tests in terms of conditional hypothesis. My questions are: In this frequentist approach, the ...
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2answers
307 views

Which are the correct sum of squares for repeated measures ? (balanced example)

Below are three ways to fit a MANOVA model in R and to extract ANOVA tables from the fitted model. The design is balanced. All methods give different results. The data are 5 groups of 4 individuals ...
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0answers
79 views

Type I ANOVA tests not depending on the order of the factors

I have a dataset with two factors A and B and the following design (contigency table showing the number of individuals for each ...
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0answers
193 views

Expectation of Sums of Squares

$\mu$=mean, $\sigma$=standard dev show that if $H_0:\mu_1=...=\mu_k$ in the normal one way classification then the Total Sum of Squares has expectation $E(T)=(N-1)\sigma^2$ the Within Groups Mean ...
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1answer
362 views

Comparison of k-means clustering output

I am trying to compare the outputs of k-means algorithm coded by me and the outputs of R's kmeans. Since the objective of the algorithm is to minimize the total ...
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1answer
294 views

Significance of the component in a multivariate linear model (“within-design”)

I'm trying to test the significance of the "component" effect in a multivariate regression model. I'm not sure what is the right way. Using R, I have tried a way with ...
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0answers
93 views

Different Mean Square partitions in an unbalanced bifactorial ANOVA (with random factor) between R and Statistica

I am trying to extract variance components for selection and chance in a bifactorial design with Generation as a fixed factor and Replicate as a random term, for early fecundity. Since I am using ...
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0answers
171 views

Sum of Squares reduction test: Convergence criteria met, but not all parameters of the model estimated

Background: I am using weighted non-linear regression to model the growth of plant organs, with dummy variables for different species. I am using a sum of squares reduction test (SSRT) to compare the ...
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1answer
4k views

How to calculate sum of squared errors (SSE or SSR) with Stata 12 software?

I have a techincal issue with Stata: I need to calculate the SSE of a regression model, but the automatic output just gives me RMSE; I need SSE of this constrained model because I have to test if this ...
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1answer
609 views

Unique variance with regression analysis

I have some questions about unique variance and hope some of you can help. For instance, let say I have 3 predictors and 1 dependent variable (DV). I ran a regression analysis with a sequence of 3 ...
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0answers
19 views

How do I get the SSE for predictions using SAS? [duplicate]

Possible Duplicate: How to get SSE for predictions using SAS? I'm trying to get the sum of squared errors for my predictions in SAS, but I'm not sure I'm doing it correctly. I'm not sure ...
0
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1answer
346 views

How to get SSE for predictions using SAS?

I have a data set with some predictors, a number of training points, and some test points. I've come up with a model that I believe most accurately represents the training data (throwing out x1 and ...
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0answers
681 views

Effect Size/Mean Squared Error from Linear Mixed-Model in R [duplicate]

I'm trying to report an effect size for a Linear Mixed-Model we've fitted in R. Right now I'm looking at reporting partial eta squared or eta squared. However, to do so I need to calculate the Sums of ...
4
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0answers
327 views

Missing cells with Type III SS

I have a linear model of the form y ~ x + z + x:z. I have unbalanced data and further have a few missing cells (data would be unbalanced even without those missing cells). My understanding of the ...
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1answer
189 views

Proper order of variables in unbalanced ANOVA

If you have unequal sample sizes in cells, then the order in which you enter model terms changes your results for sequential or Type I SS. The first variable to enter the model is allocated its ...
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1answer
2k views

Distribution of sum of squares error for linear regression?

I know that distribution of sample variance $$ \sum\frac{(X_i-\bar{X})^2}{\sigma^2}\sim \chi^2_{(n-1)} $$ $$ \sum\frac{(X_i-\bar{X})^2}{n-1}\sim \frac{\sigma^2}{n-1}\chi^2_{(n-1)} $$ It's from the ...
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2answers
1k views

Where is the shared variance between all IVs in a linear multiple regression equation?

In a linear multiple regression equation, if the beta weights reflect the contribution of each individual independent variable over and above the contribution of all the other IVs, where in the ...