Questions tagged [quadratic-form]

A quadratic form is a homogeneous polynomial of order two.

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1answer
23 views

Panel data OLS regression: plot and quadratic fitting line

I am conducting an OLS regression panel data analysis with package PLM in R. I use the following script to obtain a plot and fitting line of variables D and ...
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2answers
59 views

Assessing model fit of two models by computing t-test on coefficients in R

I am attempting to assess whether my explanatory variable of "startingpos" is best modelled via a quadratic regression model or a linear regression model. One way I can do this is to compute the ...
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1answer
289 views

Fitting a quadratic regression in R

I am trying to fit a quadratic regression model in R. Here is an example of my dataframe: ...
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1answer
28 views

Including a quadratic effect for an ordinal variable in a regression analysis

It's common for many datasets to have ordinal versions of numerical variables, such as age groups (e.g. "Under 20", "20-30", "30-40", etc.) or time groups (e.g. "Less than 15 minutes", "15-30 minutes",...
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146 views

Variance Ratio Formula

I've been trying to minimize/maximize the ratio of quadratic forms given by $$Q(c)=\frac{c^\top \Sigma c}{c^\top \text{diag}(\Sigma) c}$$ where $\Sigma$ denotes a covariance matrix of some $n$-...
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40 views

Variance of Quadratic form

From the Wikipedia In general, the variance of a quadratic form depends greatly on the distribution of ${\displaystyle \varepsilon }$ . However, if ${\displaystyle \varepsilon }$ does follow a ...
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13 views

Inflection Point in Quadratic Model

I have a panel data and I am estimating a qudaratic model with fixed effects. The following model is estimated using stata. ...
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1answer
69 views

Expectation of Quadratic Form with Two Random Vectors

Assume I have two independent $(N \times 1)$ random vectors, $\epsilon_{1} \sim N(0,\Sigma_1)$ and $\epsilon_{2} \sim N(0,\Sigma_2)$. We could assume $\Sigma_1=\Sigma_2$ for my purposes but a general ...
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37 views

Self-Study: If $x \sim \mbox N_p(\mu; V)$ and A is a matrix then how to show that $E[(x-\mu)(x-\mu)'A(x-\mu) = 0$?

If we assume the random vector $x$ to be normally distributed with $N_p(\mu; V)$ then $E[(x-\mu)(x-\mu)'(x-\mu)] = 0_p$. If I am not mistaken, this can be shown using the moment generating function of ...
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1answer
53 views

Determine statistical difference of slopes of quadratic relationship in a Poisson regression

I'm looking for a statistical or mathematical way to test the difference between two slopes. Others have asked related questions but my problem is quite particular. I'm running a Poisson regression ...
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0answers
40 views

Maximization of quotient of quadratic forms in linear regression

I would like to find maximum of the following function: $$I = \max_{a\in \mathbb{R}^p} \frac{(a'\hat{\beta})^2}{S^2a'(X'X)^{-1}a},$$ where $X$ is a design matrix and of course $Y$ is normally ...
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1answer
41 views

How to generate data such that an equation needs to hold?

Can I create or generate $\{y_i\}_{i=1}^{4}$ data set such that this equation holds $$ \sum_{i=1}^{4}\sum_{j=1}^{4}m_{ij}y_{i}y_{j}=6 $$ where $$ m=\left[ \begin{array}{cccc} 13 & 12 & 3 &...
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1answer
66 views

What is the relationship between quadraric and categorical logistic regression models?

Consider two logistic regression models Y on x, one where x appears in the model as a categorical variable, and one where x appears in the predictor as both a linear term and as a quadratic term. ...
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1answer
41 views

What is the relationship between a quadratic model and categorical model?

Using logistic explore the association between lung reactivity and risk of chronic respiratory disease. The dataset contains information on a combined measure of lung function exposure respcat ...
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1answer
133 views

Variance of quadratic form for multivariate normal distribution

This is a homework problem I’m trying to solve but I can’t seem to solve Q1b without using the theorem. I am also given the fact that $$E(y’Ay)=tr(A\Sigma)+\mu’A\mu$$ I’ve tried using the trace-...
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1answer
48 views

Regression: Why does using quadratic expressions work with linear estimators? [duplicate]

My questions is, that I see people using R´s lm() (linear regression model) with Y ~ X^2 e.g. here: Simple non-linear regression problem But I dont see how and ...
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51 views

Maximal inequality for quadratic forms as functions of matrices?

In general, an $M$-estimator is defined as a maximizer of some objective function $Q_n(\theta)$: $\hat{\theta} = \arg\max_{\theta} Q_n(\theta)$. Suppose that $q(\theta) = E Q_n(\theta)$ is uniquely ...
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23 views

Modeling approach to showing a parabolic effect is greater than that expected by scale boundaries

I have conducted an experiment where raters rate different nations on a DV. Each observation is a different nation. I calculated a mean and SD of the DV for each nation. The data we are working with ...
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64 views

Asymptotic normality of quadratic form?

Let $X$ be a $p$-dimensional vector that is asymptotically normal such that $$\sqrt{n}(X - \mu_X) \stackrel{d}\longrightarrow N(0, \Sigma)$$, and let $H$ be a random $p\times p$ symmetric matrix, ...
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1answer
331 views

How to include an interaction with a quadratic term? [closed]

I want to predict $y$ with $x_{1}$ and $x_{2}$ and I suppose that $x_{2}$ has a quadratic effect on $y$ and that there is an interaction. How to model that? I've look in previous questions but there ...
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1answer
81 views

Quadratic term of standardized predictor in logistic regression

A random intercept logistic regression is performed to assess the association between $Y$: Disease (Yes/No) and Standardized Predictor($X_1$) adjusting for control variables ($X_2$, $X_3$) based on ...
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1answer
62 views

How powerful are second order interactions?

A lot of applications in statistics and machine learning model a phenomenon by second order interactions of variables and get good results. By second order interactions I mean, for a general variable $...
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1answer
48 views

Why do we use quadratic form for random vectors? [closed]

I am studying linear regression. I have studied this in the past, but this is my first time exposing myself to the matrix form of multiple linear regression. My matrix algebra/linear algebra skills ...
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1answer
168 views

Quadratic polynomial - how to test correlation between x and y?

I have one dependent variable (fish abundance) and one independet variable (time), both continuous. I would like to test the correlation between them because I expect that the abudance changes over ...
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1answer
557 views

Mean and Variance of SSE

First, let \begin{equation} SSE = \overrightarrow{y}'(I - H) \overrightarrow{y} \end{equation} where \begin{equation} \overrightarrow{y} \sim MN(\textbf{X} \overrightarrow{\beta}, \sigma^{2}I) \\ H ...
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Test of concavity for repeated measures data

First question here. I'm trying to figure out what statistical test is appropriate for testing whether a series of data is concave or convex. Specifically, this is coming from a human subjects ...
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1answer
208 views

Interpretation of Cox Hazard Model with quadratic term

I am having trouble finding information on how to interpret coxph model hazard ratios with a quadratic term. Some of my variables are continuous count data, whereas others are continuous percentages. ...
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1answer
65 views

Quadratic model of data?

Is it possible to fit a quadratic or polynomial model with this type of data? Two inputs, input one is a temperature sensor: Input two is a valve opening on a scale from 0-100: This is a scatter ...
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1answer
270 views

Is the Quadratic Approximation of Log-Likelihood Equivalent to the Normal Approximation of the MLE?

Let $X_1, X_2, ..., X_n \sim \text{IID N}(\theta, \sigma^2)$ with $\sigma^2$ known, and let $\hat{\theta}$ be the MLE of the mean. (1) How can I show that in this case, the following is true? $$\...
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1answer
277 views

Covariance of Two Quadratic Forms

We're looking for the $\operatorname{Cov}\left[x^T A x, ~x^T B x\right]$ where $x$ is random variable and mean-centered, but not independent and $A$ and $B$ are symmetric matrices. The fundamental ...
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15 views

Sums of degenerate quadratic forms

I am searching for an analogue of the fact: let $\Sigma_1 , \Sigma_2> 0$ in $\mathbb R^{m \times m}$ and let $x,c_1, c_2 \in \mathbb R^m$ be arbitrary. Let $\Sigma_3^{-1} = \Sigma_1^{-1} + \Sigma_2^...
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27 views

Center or square first when creating a quadratic term with a grand-mean centered variable in a multilevel model

I have a standard 2-level hierarchical linear model that I’m doing (household water use is the dependent variable), and one of the level-1 (household-level) variables (average cost) includes a ...
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59 views

Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

X-Posted on math.stackexchange, apologies, though I thought this was equally relevant to both communities. I'm wondering if there exists any higher-order SVD for dimensionality reduction. Note that ...
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1answer
2k views

Interpreting a Quadratic Term in Binary Logistic Regression

Apologies in advance for my limited stats knowledge. I hope someone can help. I am trying to understand how to interpret the coefficients of both the linear and quadratic term in a binary logistic ...
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457 views

Distribution of quadratic form of multivariate normal with linear term

Suppose that $A$ is a symmetric non-random matrix and $X\sim N(\mu,\Sigma)$ and $b \in R^n$ is a non-random vector. Then what is the distribution of $$X^tAX+b^tX \quad ?$$ The distribution without ...
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67 views

Interpret Quadratic Regression with Asinh transformation

I have a regression equation that uses covariates of the following form asinh(y) = b0 + b1 asinh(x) + b2 (asinh(x))^2 + error and am wondering how to intepret the ...
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44 views

Accounting for non-linear trend in SOME samples of a linear mixed model using quadratic

I have the following data: Pine Forest Biomass ~ Age | Plot: Each black curve represents whole-plot biomass for each individual plot I want to formally examine ...
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1answer
55 views

Insignificant x, but significant x squared

I have estimated the following model to capture increasing/decreasing marginal effect of $x$ on $y$. : $y=\alpha + \beta_1x+ \beta_2x^2 +e$ where $\beta_1$ is statistically insignificant, but $\...
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1answer
366 views

Asymptotic normality of a quadratic form

Let $\mathbf{x}$ be a random vector drawn from $P$. Consider a sample $\{ \mathbf{x}_i \}_{i=1}^n \stackrel{i.i.d.}{\sim} P$. Define $\bar{\mathbf{x}}_n := \frac{1}{n} \sum_{i=1}^n \mathbf{x}_i$, and $...
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0answers
865 views

How to report results of a polynomial regression with a discrete independent variable

I have been recently trying to fit a linear model to my data. The dependent variable is continuous and the independent variable is numeric and discrete. When I first test the assumptions concerning ...
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0answers
860 views

If I have significant linear and quadratic terms, can I interpret both of these results?

I would have thought there'd be an answer to this question on here already, but I've been unable to find it, so I apologize if this is a repost. If I have a regression model of the form ...
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0answers
747 views

Mixed Effects model with quadratic term in fixed, but not in random part

Say one wishes - and believes - to fit a mixed effects model where time ($x_{ij}$ below) is included in a linear and quadratic term in the fixed part, but only linear in the random part (hence random ...
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1answer
2k views

Equivalent to ANOVA for non-linear (quadratic) relationships

I am trying to perform the equivalent of a repeated-measures ANOVA using data that have a non-linear relationship. The independent variable position runs from -20 ...
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2answers
932 views

Quadratic form and Chi-squared distribution

It's about the demostration of the quadratic forms and chi-squared distribution. Let's split the problem: We have a $n$ vector with n standardized normal distribution called $z={[z_1,z_2...z_n]}$. ...
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1answer
2k views

Linear and quadratic term interpretation in regression analysis

I'm considering the case of a factor which is detrimental to performance but including also, a quadratic term to allow for the existence of a non-linear relationship. In all cases, both linear and ...
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1answer
258 views

Proof that $\mathrm{Cov}(x^TAx,x^TBx) = 2 \mathrm{Tr}(A \Sigma B \Sigma) + 4 \mu^TA \Sigma B \mu$

Suppose $\vec x \sim N(\vec \mu, \Sigma)$ is multivariate normal. I want to see that $\mathrm{Cov}(\vec x^TA\vec x,\vec x^TB\vec x) = 2 \mathrm{Tr}(A \Sigma B \Sigma) + 4 \vec \mu^TA \Sigma B \vec \...
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40 views

If $\vec y \sim N(\vec 0,\Sigma)$, what is the distribution of $y^TAy$, when $A$ is indefinite?

Suppose $\vec y \sim N(\vec 0,\Sigma)$ with $\Sigma$ singular. Is the distribution of $y^TAy$ known in the case that $A$ is indefinite?
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Mix Design: Failure to predict response

I have performed a mixture design containing 3 terms and I have carried out 6 experiments needed to adjust a quadratic model. It fitted the data quite well: ...
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0answers
182 views

Quantiles of linear combination of independent $\chi^2_1$ random variables [duplicate]

I want to work out the quantiles of a linear combiation of chi square random variables. Suppose $\lambda_i \in \mathbb{R}$ for all $i \in \{1,2,\cdots,n\}$ and $Z = \sum_{i = 1}^n \lambda_iX_i$ and $...
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1answer
185 views

Calculate average slope in a quadratic

Let's say I have crop yield on y-axis and a measure of heat-stress on x-axis According to this graph, as heat increases, the yield also increases and after reaching a maximum (at heat = 0.67), the ...