Calculating Standard Error Of Linear Regression
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the estimate from a scatter plot Compute the standard error of the estimate based on errors of prediction Compute the standard error using Pearson's correlation Estimate the standard error of the estimate based on a sample Figure 1 shows two regression examples. You can see that standard error multiple linear regression in Graph A, the points are closer to the line than they are in Graph B. standard error simple linear regression Therefore, the predictions in Graph A are more accurate than in Graph B. Figure 1. Regressions differing in accuracy of prediction. The standard standard error linear regression excel error of the estimate is a measure of the accuracy of predictions. Recall that the regression line is the line that minimizes the sum of squared deviations of prediction (also called the sum of squares error). The standard error standard error linear regression slope of the estimate is closely related to this quantity and is defined below: where σest is the standard error of the estimate, Y is an actual score, Y' is a predicted score, and N is the number of pairs of scores. The numerator is the sum of squared differences between the actual scores and the predicted scores. Note the similarity of the formula for σest to the formula for σ.  It turns out that σest is the
Standard Error Linear Regression In R
standard deviation of the errors of prediction (each Y - Y' is an error of prediction). Assume the data in Table 1 are the data from a population of five X, Y pairs. Table 1. Example data. X Y Y' Y-Y' (Y-Y')2 1.00 1.00 1.210 -0.210 0.044 2.00 2.00 1.635 0.365 0.133 3.00 1.30 2.060 -0.760 0.578 4.00 3.75 2.485 1.265 1.600 5.00 2.25 2.910 -0.660 0.436 Sum 15.00 10.30 10.30 0.000 2.791 The last column shows that the sum of the squared errors of prediction is 2.791. Therefore, the standard error of the estimate is There is a version of the formula for the standard error in terms of Pearson's correlation: where ρ is the population value of Pearson's correlation and SSY is For the data in Table 1, μy = 2.06, SSY = 4.597 and ρ= 0.6268. Therefore, which is the same value computed previously. Similar formulas are used when the standard error of the estimate is computed from a sample rather than a population. The only difference is that the denominator is N-2 rather than N. The reason N-2 is used rather than N-1 is that two parameters (the slope and the intercept) were estimated in order to estimate the sum of squares. Formulas for a sample comparable to the ones for a population are shown below. Please answer the questions: feedback
1: descriptive analysis · Beer sales vs. price, part 2: fitting a simple model · Beer sales vs. price, part 3: transformations of variables · Beer sales vs. price, part 4: additional predictors · NC natural gas consumption vs. temperature
Standard Error Linear Regression Spss
What to look for in regression output What's a good value for R-squared? What's standard error linear regression equation the bottom line? How to compare models Testing the assumptions of linear regression Additional notes on regression analysis Stepwise and all-possible-regressions standard error linear regression matlab Excel file with simple regression formulas Excel file with regression formulas in matrix form If you are a PC Excel user, you must check this out: RegressIt: free Excel add-in for linear regression and multivariate http://onlinestatbook.com/2/regression/accuracy.html data analysis Mathematics of simple regression Review of the mean model Formulas for the slope and intercept of a simple regression model Formulas for R-squared and standard error of the regression Formulas for standard errors and confidence limits for means and forecasts Take-aways Review of the mean model To set the stage for discussing the formulas used to fit a simple (one-variable) regression model, let′s briefly review the formulas http://people.duke.edu/~rnau/mathreg.htm for the mean model, which can be considered as a constant-only (zero-variable) regression model. You can use regression software to fit this model and produce all of the standard table and chart output by merely not selecting any independent variables. R-squared will be zero in this case, because the mean model does not explain any of the variance in the dependent variable: it merely measures it. The forecasting equation of the mean model is: ...where b0 is the sample mean: The sample mean has the (non-obvious) property that it is the value around which the mean squared deviation of the data is minimized, and the same least-squares criterion will be used later to estimate the "mean effect" of an independent variable. The error that the mean model makes for observation t is therefore the deviation of Y from its historical average value: The standard error of the model, denoted by s, is our estimate of the standard deviation of the noise in Y (the variation in it that is considered unexplainable). Smaller is better, other things being equal: we want the model to explain as much of the variation as possible. In the mean model, the standard error of the model is just is the sample standard devi
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow http://stats.stackexchange.com/questions/85943/how-to-derive-the-standard-error-of-linear-regression-coefficient the company Business Learn more about hiring developers or posting ads with us Cross http://stats.stackexchange.com/questions/33260/how-to-calculate-the-interaction-standard-error-of-a-linear-regression-model-in Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and standard error rise to the top How to derive the standard error of linear regression coefficient up vote 2 down vote favorite 3 For this univariate linear regression model $$y_i = \beta_0 + \beta_1x_i+\epsilon_i$$ given data set $D=\{(x_1,y_1),...,(x_n,y_n)\}$, the coefficient estimates are $$\hat\beta_1=\frac{\sum_ix_iy_i-n\bar x\bar y}{n\bar x^2-\sum_ix_i^2}$$ $$\hat\beta_0=\bar y - \hat\beta_1\bar x$$ Here is my question, according to the book and Wikipedia, the standard error of $\hat\beta_1$ is $$s_{\hat\beta_1}=\sqrt{\frac{\sum_i\hat\epsilon_i^2}{(n-2)\sum_i(x_i-\bar x)^2}}$$ How and why? standard-error inferential-statistics share|improve standard error linear this question edited Mar 6 '15 at 14:38 Christoph Hanck 9,13332149 asked Feb 9 '14 at 9:11 loganecolss 5531926 stats.stackexchange.com/questions/44838/… –ocram Feb 9 '14 at 9:14 @ocram, thanks, but I'm not quite capable of handling matrix stuff, I'll try. –loganecolss Feb 9 '14 at 9:20 1 @ocram, I've already understand how it comes. But still a question: in my post, the standard error has $(n-2)$, where according to your answer, it doesn't, why? –loganecolss Feb 9 '14 at 9:40 add a comment| 1 Answer 1 active oldest votes up vote 7 down vote accepted 3rd comment above: I've already understand how it comes. But still a question: in my post, the standard error has (n−2), where according to your answer, it doesn't, why? In my post, it is found that $$ \widehat{\text{se}}(\hat{b}) = \sqrt{\frac{n \hat{\sigma}^2}{n\sum x_i^2 - (\sum x_i)^2}}. $$ The denominator can be written as $$ n \sum_i (x_i - \bar{x})^2 $$ Thus, $$ \widehat{\text{se}}(\hat{b}) = \sqrt{\frac{\hat{\sigma}^2}{\sum_i (x_i - \bar{x})^2}} $$ With $$ \hat{\sigma}^2 = \frac{1}{n-2} \sum_i \hat{\epsilon}_i^2 $$ i.e. the Mean Square Error (MSE) in the ANOVA table, we end up with your expression for $\widehat{\text{se}}(\hat{b})$. The $n-2$ term accounts for the loss of 2 degrees of freedom in the estimation of the intercept and the slope. share|improve this answ
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Cross Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top How to calculate the interaction standard error of a linear regression model in R? up vote 6 down vote favorite 2 I have the following model and want to make a table with the interpretation of the interaction effects as suggested by Bambor and Clark in this paper. However, I have no idea on how to calculate the $cov(\hat{\beta_1}, \hat{\beta_5})$ in the formula. The model: reg <- lm( log(Y) ~ as.factor(X1) + as.factor(X2) + log(as.numeric(X3)) + log(as.numeric(X4)) + as.factor(X1)*as.factor(X2), data=DB) The results: Residuals: Min 1Q Median 3Q Max -5.1091 -0.3036 0.0294 0.3396 3.6537 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.08848 0.09523 -0.929 0.3531 as.factor(X1)1 -0.12795 0.06227 -2.055 0.0402 * as.factor(X2)1 0.05666 0.06694 0.846 0.3976 log(as.numeric(X3)) 0.03602 0.02121 1.699 0.0898 . log(as.numeric(X4)) 0.97546 0.02671 36.514 <2e-16 *** as.factor(X1)1:as.factor(X2)1 0.10733 0.11790 0.910 0.3629 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.6729 on 795 degrees of freedom Multiple R-squared: 0.9161, Adjusted R-squared: 0.9155 F-statistic: 1735 on 5 and 795 DF, p-value: < 2.2e-16 The formula: $\sqrt{var(\hat{\beta_1}) + var(\hat{\beta_5}) + 2cov(\hat{\beta_1}\hat{\beta_5})}$ r regression interaction interpretation share|improve this question edited Jul 28 '12 at 17:56 asked Jul 28 '12 at 14:59 Davi Moreira 13316 1 Ah, sorry, didn't fully answer your question. $cov(\hat{\beta}_1,\hat{\beta}_2)$ would be the entry in the matrix with column name, in your case, as.factor(X1)1 row name as.factor(X2)1, assuming what you wished to find the covariance for were the coefficients with values -0.12795 and 0.05666, a