Calculating Standard Error - Regression Analysis
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it comes to determining how well a linear model fits the data. However, I've stated previously that R-squared is overrated. Is there a different goodness-of-fit statistic that can be more helpful? You bet! Today, I’ll highlight how to calculate standard error of regression coefficient a sorely underappreciated regression statistic: S, or the standard error of the regression. S provides how to calculate standard error of regression in excel important information that R-squared does not. What is the Standard Error of the Regression (S)? S becomes smaller when the data points how to calculate standard error of regression slope are closer to the line. In the regression output for Minitab statistical software, you can find S in the Summary of Model section, right next to R-squared. Both statistics provide an overall measure of how well standard error linear regression the model fits the data. S is known both as the standard error of the regression and as the standard error of the estimate. S represents the average distance that the observed values fall from the regression line. Conveniently, it tells you how wrong the regression model is on average using the units of the response variable. Smaller values are better because it indicates that the observations are closer to the fitted line.
Standard Error Multiple Regression
The fitted line plot shown above is from my post where I use BMI to predict body fat percentage. S is 3.53399, which tells us that the average distance of the data points from the fitted line is about 3.5% body fat. Unlike R-squared, you can use the standard error of the regression to assess the precision of the predictions. Approximately 95% of the observations should fall within plus/minus 2*standard error of the regression from the regression line, which is also a quick approximation of a 95% prediction interval. For the BMI example, about 95% of the observations should fall within plus/minus 7% of the fitted line, which is a close match for the prediction interval. Why I Like the Standard Error of the Regression (S) In many cases, I prefer the standard error of the regression over R-squared. I love the practical, intuitiveness of using the natural units of the response variable. And, if I need precise predictions, I can quickly check S to assess the precision. Conversely, the unit-less R-squared doesn’t provide an intuitive feel for how close the predicted values are to the observed values. Further, as I detailed here, R-squared is relevant mainly when you need precise predictions. However, you can’t use R-squared to assess the precision, which ultimately leaves it unhelpful. To il
1: descriptive analysis · Beer sales vs. price, part 2: fitting a simple model · Beer sales vs. price, part 3: transformations of variables ·
Standard Error Of Regression Formula
Beer sales vs. price, part 4: additional predictors · NC natural confidence interval regression analysis gas consumption vs. temperature What to look for in regression output What's a good value for R-squared? t test regression analysis What's the bottom line? How to compare models Testing the assumptions of linear regression Additional notes on regression analysis Stepwise and all-possible-regressions Excel file with simple regression http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-to-interpret-s-the-standard-error-of-the-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 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 http://people.duke.edu/~rnau/mathreg.htm 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 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 i
the ANOVA table (often this is skipped). Interpreting the regression coefficients table. Confidence intervals for the slope parameters. Testing for statistical significance of coefficients Testing hypothesis on a slope parameter. Testing overall significance of the regressors. Predicting y http://cameron.econ.ucdavis.edu/excel/ex61multipleregression.html given values of regressors. Excel limitations. There is little extra to know beyond regression with one explanatory variable. The main addition is the F-test for overall fit. MULTIPLE REGRESSION USING THE DATA ANALYSIS ADD-IN This requires http://stats.stackexchange.com/questions/44838/how-are-the-standard-errors-of-coefficients-calculated-in-a-regression the Data Analysis Add-in: see Excel 2007: Access and Activating the Data Analysis Add-in The data used are in carsdata.xls We then create a new variable in cells C2:C6, cubed household size as a regressor. standard error Then in cell C1 give the the heading CUBED HH SIZE. (It turns out that for the se data squared HH SIZE has a coefficient of exactly 0.0 the cube is used). The spreadsheet cells A1:C6 should look like: We have regression with an intercept and the regressors HH SIZE and CUBED HH SIZE The population regression model is: y = β1 + β2 x2 + β3 x3 + u It standard error of is assumed that the error u is independent with constant variance (homoskedastic) - see EXCEL LIMITATIONS at the bottom. We wish to estimate the regression line: y = b1 + b2 x2 + b3 x3 We do this using the Data analysis Add-in and Regression. The only change over one-variable regression is to include more than one column in the Input X Range. Note, however, that the regressors need to be in contiguous columns (here columns B and C). If this is not the case in the original data, then columns need to be copied to get the regressors in contiguous columns. Hitting OK we obtain The regression output has three components: Regression statistics table ANOVA table Regression coefficients table. INTERPRET REGRESSION STATISTICS TABLE This is the following output. Of greatest interest is R Square. Explanation Multiple R 0.895828 R = square root of R2 R Square 0.802508 R2 Adjusted R Square 0.605016 Adjusted R2 used if more than one x variable Standard Error 0.444401 This is the sample estimate of the standard deviation of the error u Observations 5 Number of observations used in the regression (n) The above gives the overall goodness-of-fit measures: R2 = 0.8025 Correlation between y and y-hat is 0.8958 (when squared gives 0.8025). Ad
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 are the standard errors of coefficients calculated in a regression? up vote 53 down vote favorite 43 For my own understanding, I am interested in manually replicating the calculation of the standard errors of estimated coefficients as, for example, come with the output of the lm() function in R, but haven't been able to pin it down. What is the formula / implementation used? r regression standard-error lm share|improve this question edited Aug 2 '13 at 15:20 gung 73.6k19160307 asked Dec 1 '12 at 10:16 ako 368146 good question, many people know the regression from linear algebra point of view, where you solve the linear equation $X'X\beta=X'y$ and get the answer for beta. Not clear why we have standard error and assumption behind it. –hxd1011 Jul 19 at 13:42 add a comment| 3 Answers 3 active oldest votes up vote 68 down vote accepted The linear model is written as $$ \left| \begin{array}{l} \mathbf{y} = \mathbf{X} \mathbf{\beta} + \mathbf{\epsilon} \\ \mathbf{\epsilon} \sim N(0, \sigma^2 \mathbf{I}), \end{array} \right.$$ where $\mathbf{y}$ denotes the vector of responses, $\mathbf{\beta}$ is the vector of fixed effects parameters, $\mathbf{X}$ is the corresponding design matrix whose columns are the values of the explanatory variables, and $\mathbf{\epsilon}$ is the vector of random errors. It is well known that an estimate of $\mathbf{\beta}$ is given by (refer, e.g., to the wikipedia article) $$\hat{\mathbf{\beta}} = (\mathbf{X}^{\prime} \mathbf{X})^{-1} \mathbf{X}^{\prime} \mathbf{y}.$$ Hence $$ \textrm{Var}(\hat{\mathbf{\beta}}) = (\mathbf{X}^{\prime} \mathbf{X})^{-1} \mathbf{X}^{\prime} \;\si