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2
votes
2answers
51 views

K-means: Why minimizing WCSS is maximizing Distance between clusters?

From a conceptual and algorithmic standpoint, I understand how K-means works. However, from a mathematical standpoint, I don't understand why minimizing the WCSS (within-cluster sums of squares) will ...
1
vote
0answers
11 views

sadists package/ Sum of (non-central) chi-squares to a power

I try use "sadists" package in R to compute quantiles and probabilities on sum of non-central chi-squares distribution, but there are some issues in this package. I give an example: wts = ...
0
votes
1answer
19 views

How does ANCOVA increase the variance explained by the categorical predictor?

I'm reading Field (2013), in which there is an ANCOVA example with a categorical predictor Viagra, a continuous covariate, and a continuous outcome variable libido. I understand that in ANCOVA we ...
3
votes
2answers
81 views

Difference of two squared normal dependent variables

I need to find the distribution of the random variable $Z$ $Z = \frac{(X - \mu_0)^2}{\sigma_0^2} - \frac{(X - \mu_1)^2}{\sigma_1^2}$, where $X \sim \mathcal{N}(\mu_0, \sigma_0)$. We can find the ...
0
votes
0answers
17 views

Relative Proportion of Variance for each Factor (ANOVA)

I'd like to calculate how much of the variance is related to each of the input factors. For example (in this case some R-code), if one wants to calculate how much of the "Magic Effect" of a magic ...
1
vote
1answer
43 views

A proof of total sum of squares being equal to within-cluster sum of squares and between cluster sum of squares? [duplicate]

In cluster analysis I have frequently encountered a statement that the total sum of squares $\sum\limits_{i = 1}^n {{{({x_i} - \overline x )}^2}} $ being equal to within-cluster sum of squares ...
1
vote
1answer
36 views

Mean squared error versus Least squared error, which one to compare datasets?

I have 3 datasets of the same system. But for the first one, I have 21 measurements. For the second and the third one I have only 9 measurements. Now I made a model using these 3 datasets (so 3 ...
1
vote
0answers
43 views

Type III SS, anova - comparison to residuals

Apologies if this topic seems to have been beaten to death, but I couldn't find an exact duplicate. Take this data in R: ...
2
votes
2answers
142 views

Mixed Model Type-III Sums of Squares- R vs SPSS

The age old question of comparing sums of squares (SS) between programs has reared its ugly head again. I am trying to replicate output in SPSS, that was computed using Type 3 Sums of Squares, in R. ...
1
vote
1answer
170 views

Why report r-squared in Instrumental Variables Estimation?

I mean the the R-squared calculated such as in $R^2=1-\frac{RSS}{TSS}$ when you use the $RSS$ from the original structural model and not recalculation that you should do in order to do an F test. With ...
0
votes
1answer
68 views

Why does adding more terms into a linear model always increase the r-squared value?

Many statistics textbooks state that adding more terms into a linear model always reduces the sum of squares and in turn increases the r-squared value. This has led to the use of the adjusted ...
1
vote
0answers
66 views

Regression sum of squares (SSR) when $\beta_0 = 0$ in multiple linear regression?

I hope this is not a duplicate but I cannot find the answer to this question. In a linear model $$Y_i = \beta_1 X_{i,1} + \dots + \beta_{p-1} X_{i,p-1} + \varepsilon_i, \qquad i = 1, \ldots, n$$ with ...
2
votes
1answer
103 views

Sum of Square decomposition

Question about the Total, Explained, and Residual Sum of Squares. I am in the simple linear regression model. Could you help me clarify why the residual sum of squares (SSE where E stands for errors) ...
7
votes
4answers
802 views

square things in statistics- generalized rationale [duplicate]

Why do you square things in stats? I have run across this a lot, in both data mining and statistics classes, but no one has ever been able to give me an answer. One specific example is when summing ...
0
votes
1answer
37 views

Vectorise Within-Groups Sum of Squares in R

I've got a multivariate dataset (p=2) that I'm trying to calculate the W matrix for use in canonical variates analysis If each $x_{kj}$ is the jth observational unit from the kth group, and ...
4
votes
1answer
198 views

Why does the SSR have 1 degree of freedom in simple linear regression?

I understand degrees of freedom as the number of things that can independently change. And typically, in coming up with the degrees of freedom, if you have n terms, then you just subtract out the ...
0
votes
1answer
30 views

ANOVA in R: How MSresiduals are estimated when using Error(interaction)?

I have checked the existing answers, but I found no answer to my question. It is about choosing the right ANOVA model in R and how the MSresiduals are estimated. I have an experiment with two ...
1
vote
1answer
212 views

Orthogonal contrasts for four levels - are Helmert contrasts orthogonal?

I am planning to do a 2-way mixed ANOVA with the within-subjects factor having four levels. Since the exact data do not matter, I hope to suffice by only providing a general question. On my ...
3
votes
1answer
110 views

What exactly does a Type III test do?

I'm having trouble understanding what exactly Type III test statistic does. Here is what I got from my book: "Type III" tests test for the significance of each explanatory variable, under the ...
1
vote
1answer
121 views

Deriving Regression Sum of Squares (SSR)

In a simple linear regression $SSR = \sum(\hat{y} - \bar{y})^2$ $SSXY = \sum(x - \bar{x})(y-\bar{y})$ $SSX = \sum(x - \bar{x})^2$ $SSR$ can be computed by dividing $SSXY^2$ by $SSX$. Namely, $SSR ...
1
vote
0answers
103 views

choosing $β_0$ and $β_1$ to minimize the residual sum of squares

I'm reading a book called An Introduction to Statistical Learning: with Applications in R, and I have a question in regards to the material inside. I understand that we can find the residual sum of ...
1
vote
0answers
53 views

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, ...
2
votes
0answers
82 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\\ ...
1
vote
1answer
35 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 ...
1
vote
1answer
52 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? ...
0
votes
1answer
169 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: ...
1
vote
1answer
339 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 = ...
1
vote
1answer
110 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 ...
1
vote
1answer
2k views
1
vote
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 ...
0
votes
1answer
3k views

Calculating sum of squares between groups

I have carried out this ANOVA: ...
3
votes
0answers
149 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 + ...
1
vote
0answers
27 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 ...
0
votes
1answer
47 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, ...
0
votes
1answer
967 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) ...
1
vote
2answers
767 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 ...
8
votes
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 ...
1
vote
0answers
145 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?
1
vote
0answers
1k 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 ...
3
votes
1answer
222 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 ...
0
votes
0answers
149 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 ...
6
votes
2answers
10k 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?
0
votes
1answer
174 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 ...
0
votes
1answer
329 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 ...
2
votes
1answer
76 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 ...
11
votes
3answers
8k 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 ...
1
vote
0answers
77 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 ...
1
vote
2answers
2k 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 ...
1
vote
0answers
132 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). ...
8
votes
1answer
555 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 ...