Standard error of the mean of several values of y predicted from a multiple regression

I have a multiple regression equation that predicts a trait of interest ($y$) from two measured traits ($x_1$ and $x_2$). I want to measure $x_1$ and $x_2$ for $k$ individuals of a certain plant species, and use this regression to estimate the mean and standard error of $y$ for this species. I know the standard … Read more

method to compare means including standard errors

Let’s say you want to compare the means of 25 groups and you also want to consider the confidence interval of the mean. There is an issue because some group will have higher observations than other groups . Below is some sample data that represents 5 of the 25 groups. You can see for group … Read more

How did researchers calculate the Hazard ratio?

I’m looking at a pdf document and can get the result of the HR. It is actually similiar to calculating the relative risk if I’m right. How do I get the 1.45 HR or 1.20 HR values based on the 3 rows above (ppis users, h2 users, nonusers). I believe I need more parameters which … Read more

Numerical issues in Maximum Likelihood Estimation

I am estimating the following noise model: \begin{equation} n \sim \mathcal N\left(\mu(x,y),\sigma(x,y)^2\right)\in\mathbb R \end{equation} \begin{equation} \begin{cases} \mu(x,y) = k_1+k_2x+k_3x^2+k_4y+k_5y^2 \\ \sigma(x,y) = k_6+k_7x+k_8x^2+k_9y+k_{10}y^2 \\ \end{cases} \end{equation} where $x\in[0,3]$ and $y\in[0,\pi/2]$ (thus, scaling does not immediately seem to be an issue). The sample size $\{\hat n_i, \hat x_i, \hat y_i\}_{i=1}^N$ has size $N=10981$. Here are some … Read more

Calculation of confidence interval for relative difference using bootstrap

I want to calculate confidence interval for relative difference using bootstrap. I have two sample $s_a$ and $s_b$ from two difference populations $P_A$ and $P_B$ respectively. I used relative difference to compare mean value of two samples from the following formula: $ relative\_difference = \frac{m_a -m_b}{\frac{m_a+m_b}{2}}$ which $m_a$ and $m_b$ are the mean of samples … Read more

Difference between estimation and prediction in simple linear regression model?

Here is what my notes say about estimation and prediction: Estimating the conditional mean We need to estimate the conditional mean $\beta_0+\beta_1x_0$ at a value $x_0$, so we use $\hat{Y_0}=\hat{\beta_0}+\hat{\beta_1}x_0$ as a natural estimator. Here we get $$ \hat{Y_0} \sim N\left(\beta_0+\beta_1x_0,\sigma^2h_{00}\right) \,\,\,\,\,\,\,\,\,\,\,\ \text{where} \,\,\,\,\,\,\,\,\,\,\,\ h_{00} = \frac{1}{n}+\frac{(x_0-\bar{x})^2}{(n-1)s_x^2} $$ with a confidence interval for $E(Y_0) =\beta_0+\beta_1x_0$ … Read more

Finding the one-sided confidence interval for means

I don’t know where I’m making a mistake on the solution for the question. The question is as follows: Suppose that (x1,…,xn) is a sample from an N(μ,σ2) distribution, μ unknown but σ2 is known. Suppose we want to find an interval C(x1,…,xn)=(−∞,u(x1,…,xn)) that covers the interval (−∞,μ) with probability at least γ. Then we … Read more

Testing for a significant difference between 2 groups

Question 1 Two equally sized patches of the night sky are examined: Patch A contains 100 stars Patch B contains 110 stars Is there a significant difference between these two patches of night sky? i.e., is one patch likely to contain a star cluster? Question 2 Traffic to a site is examined in two time … Read more

Confidence vs prediction intervals

I try understand the difference between prediction and confidence intervals. Say we want to predict the winning time of the 100 m race in the Olympics 2020. We have fitted a linear regression model of times from 1900 to 2016 and we want an associated 95 % confidence interval of our prediction time. Would we … Read more