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Linear Regression Analysis with Minitab Fitted Line Plot Tool Dr. Doerre Data Analyses and Statistical Concepts in Biotechnology FSU Math 924 A short tutorial TOOL 1: Select Stat -> Regression -> Fitted Line Plot Minitab offers two different tools for linear regression: TOOL 2: Select Stat -> Regression -> Fit Regression Model For simple (one predictor variable) linear (and quadratic or cubic non-linear) regression, the Fitted Line Plot tool is fully sufficient. Since it has a lot fewer options, it is easier to use. The Fit Regression Model tool is mostly needed for multiple linear regression. When to use which tool? What’s the difference? The Fitted Line Plot tool provides options to output all the results needed to evaluate a linear regression. In addition, it generates the fitted line plot, with the data points, the fitted line as well as bands for the confidence interval and the prediction interval. The Fit Regression Model tool does not display the fitted line plot. However, it provides the p values and confidence intervals for the line coefficients, which the Fitted Line Plot doesn’t. TIP: Look on Blackboard for the following Excel file, which compares all the output options of these tools with the Excel Regression Data Analysis tool: Comparison of Output options Minitab Fitted Line Plot, Fit Regression Model, Excel Regression Tool.xls Where to find the Fitted Line Plot Tool Select Stat -> Regression -> Fitted Line Plot Fitted Line Plot Main Menu In addition to linear regression, it offers quadratic and cubic regression. The Main Menu has 3 buttons for more selections: “Graphs” “Options” and “Storage” These will be explained in the 3 next slides. Here is where you tell Minitab where to find your data. Make sure you properly assign the Response (Y) and Predictor (X) variable! Regression will turn out a very different result if you confuse the two (even though the correlation coefficient would be the same! Graphs Sub-Menu I recommend to get all the graphs in one file by checking “Four in one”. Recommended options: Display a band for the confidence interval (band including the range of all possible straight lines within your confidence level – default 95\%). Display a band for the prediction interval: (tell you how far off could a single point be to still fall within the statistics). You will see how these look like in an output sample further below. Options sub-menu – Even though it does not say graph or plot, it only provides options for the fitted line. Here you can select log scale axes Here you can give your plot a title. These two very useful options are explained below Storage sub-menu – It looks like this only refers to numbers that you want to store in a Minitab worksheet, as opposed to have them displayed in the Sessions window. However, it turns out that these numbers will not be calculated or displayed, if you don’t select to store them in a worksheet. *Slope anBackground Motulsky example for correlation and linare regression (chapters 22 and 23) lipids and insulin sensitivity Experiment: select random, healthy, male subjects. infuse insulin at a standard rate. infuse glucose to maintain blood glucose level constant This determines insulin sensitivity as follows: insulin triggers cells to take up glucose. the higher someones insulin sensitivity, the more glucose is taken up at the same insulin level. Thus, the speed of glucose uptake (in mg/m2/min) is an indirect measure for insulin sensitivity. Total glucose uptake (as measured from declining levels in the blood) is also dependent on the size and weight of an individual. Therefore, the speed of glucose uptake is standardized. This is done not by weight but by body surface (in m2). Determine the fatty acid composition of muscle cells (cell membranes) from biopsies. Specifically, analyze the proportion of C20-C22 polyunsaturated fatty acids within all fatty acids. Raw data \%C20-22 fatty acids Insulin sensitivity polyunsaturated [mg/m2/min] 17.9 250 18.3 220 18.3 145 18.4 115 18.4 230 20.2 200 20.3 330 21.8 400 21.9 370 22.1 260 23.1 270 24.2 530 24.4 375 Correlation between insulin sensitivity and fatty acid content 17.899999999999999 18.3 18.3 18.399999999999999 18.399999999999999 20.2 20.3 21.8 21.9 22.1 23.1 24.2 24.4 250 220 145 115 230 200 330 400 370 260 270 530 375 \%C20-22 Fatty Acids Insulin sensitivity [mg/m2/min] lines of best fit Correlation between insulin sensitivity and fatty acid content 17.899999999999999 18.3 18.3 18.399999999999999 18.399999999999999 20.2 20.3 21.8 21.9 22.1 23.1 24.2 24.4 250 220 145 115 230 200 330 400 370 260 270 530 375 \%C20-22 Fatty Acids Insulin sensitivity [mg/m2/min] calculation of R, m, and b \%C20-22 Insulin sens. fatty acids [mg/m2/min] distance to mean Product of distances (distance of x)2 X Y (for slope calculation) 17.9 250 -2.815 -34.231 96.373 7.926 18.3 220 -2.415 -64.231 155.142 5.834 18.3 145 -2.415 -139.231 336.296 5.834 18.4 115 -2.315 -169.231 391.834 5.361 18.4 230 -2.315 -54.231 125.565 5.361 20.2 200 -0.515 -84.231 43.411 0.266 20.3 330 -0.415 45.769 -19.012 0.173 21.8 400 1.085 115.769 125.565 1.176 21.9 370 1.185 85.769 101.604 1.403 22.1 260 1.385 -24.231 -33.550 1.917 23.1 270 2.385 -14.231 -33.935 5.686 24.2 530 3.485 245.769 856.411 12.143 24.4 375 3.685 90.769 334.450 13.576 Average 20.715 284.231 2480.154 66.657 Std.Dev.S 2.357 113.887 Std.Dev.P 2.264 109.419 r r2 0.834 0.6958 Motulsky r2 0.5929 0.7700025428 0.5929 OK. So they use the sample standard deviation and not the population standard deviation! Calculation of slope (I call it m the formla calls it b) and intercept (I call it b, the formula calls it a) 2480.1538461538 Sheet1 OUTPUTS AND OUTPUT OPTIONS FOR REGRESSION ANALYSIS Comparison between minitab Fitted Line Plot, minitab Fit Regression Model, and EXCEL Regression Data Analysis Tool minitab Fitted Line Plot minitab Fit Regression Model EXCEL Regression Tool where to find where to find where to find MODEL SUMMARY/Regression Statistics Pearson R, R, or multiple R no no ü default R square ü default ü default (or Results) ü default Confidence interval for R no no no calculate it yourself, see how to do it in file: Adj R square ü default ü default (or Results) ü default Manual regression calculation of Motulsky example (ch. 22-23).xls Standard Error (of the residuals) or S ü default ü default (or Results) ü default (Motulsky shows a confidence interval fo R in table 23.1) (= root mean square error) (so maybe Graphpad Prism calculates it) # Observations no no ü default ANALYSIS OF VARIANCE or ANOVA Sums of Squares (SS) Regression ü default ü default (or Results) ü default Residual or Error ü default ü default (or Results) ü default Total ü default ü default (or Results) ü default Degrees of Freedom or DF Regression ü default ü default (or Results) ü default Residual or Error ü default ü default (or Results) ü default Total ü default ü default (or Results) ü default Mean Squares (MS) Regression ü default ü default (or Results) ü default Residual or Error ü default ü default (or Results) ü default Total ü default ü default (or Results) ü default F statistic or F ü default ü default (or Results) ü default p-value or P (or significance) ü default ü default (or Results) ü default REGRESSION EQUATION ü default ü default (or Results) no COEFFICIENTS TABLE Intercept ü default ü default (or Results) ü default Slope ü default ü default (or Results) ü default (often named after X variable) Intercept Standard Error no ü default (or Results) ü default Slope Standard Error no ü default (or Results) ü default Intercept t Stat no ü default (or Results) ü default Slope t Stat no ü default (or Results) ü default Intercept P-value no ü default (or Results) ü default Slope P-value no ü default (or Results) ü default Intercept Lower 95\% CI no (but graph has it) no ü default Slope Lower 95\% CI no (but graph has it) no ü default Intercept Upper 95\% CI no (but graph has it) no ü default Slope Upper 95\% CI no (but graph has it) no ü default RESIDUALS OUTPUT (table with all the numbers) Residuals ü Storage ü Storage ü default Predicted Y (or FITS) ü Storage ü Storage ü default Standard Residuals ü Storage ü Storage ü default RESIDUAL PLOTS Normal Probability Plot ü Graphs ü Graphs ü main menu Residuals vs. x or fitted y ü Graphs ü Graphs ü main menu Histogram ü Graphs ü GraphDATA & Analysis Instructions EXCELS REGRESSION ANALYSIS TOOL Found in the menu for Excels Data Analysis Tool (if you installed that) Its free and already on your computer (if you have Excel 2013 and beyond on a PC) SAMLE DATA* Insulin sensitivity \%C20-22 fatty acids STEP 1 [mg/m2/min] polyunsaturated Open the dialog box for the Data analysis tool, click on the DATA tab and then on the icon for the Data Analysis tool all the way to the right. Y X 250 17.9 220 18.3 145 18.3 115 18.4 230 18.4 200 20.2 330 20.3 400 21.8 370 21.9 STEP 2 260 22.1 Scroll down to Regression and hit OK 270 23.1 530 24.2 375 24.4 *Data from Motulsky textbook, p. 145 STEP 3 Enter the columns for the Y data and for the X data. Check off as many of the outputs as you like. Remember to pick your DEPENDENT (outcome) variable as Y and your INDEPENDENT (predictor) variable as X Otherwise the regression analysis does not make sense! TIP: Since the output is quite comprehensive, I recommend that you select a new worksheet ply (or tab) for the results. I selected new worksheet ply. Once Excel put the data in there, I renamed the new ply Regression Results. You can see that ply / tab below. Click on it to get an explanation of all the result outputs. I selected to obtain 3 types of plots: Residual Plots, Line Fit Plots, and Normal Probability Plots. They originally appeared in the same tab. However, in order to have more space to add explanations, I moved them to a new tab that I named Regression Plots. Regression Results SUMMARY OUTPUT Regression Statistics Multiple R 0.7700025428 Pearson correlation coefficient R R Square 0.592903916 Square of R Adjusted R Square 0.5558951811 ignore - only relevant for multiple regression analysis Standard Error 75.8954867325 of residuals Observations 13 Hypothesis test for regression Excel uses a hypothesis test for the analysis of two variances (ANOVA) Null hypothesis: regression variance ≤ residual variance (MSregression ≤ MSresiduals) Alternative hypothesis: regression variance > residual variance Test statistic: F F= MS regression / MS residuals (you can check this out below) ANOVA df SS MS F Significance F Regression 1 92280.9337222659 92280.9337222659 16.020648028 0.0020770122 this is the p-value for the regression Residual 11 63361.3739700418 5760.1249063674 Total 12 155642.307692308 This table is the coefficients table Excel does not provide the formula for the fitted stright line. Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Lower 95.0\% Upper 95.0\% You can either take it from a scatter plot you made (should be the same) Intercept -486.5419945992 193.7160213841 -2.5116249607 0.0289028264 -912.9080829377 -60.1759062607 -912.9080829377 -60.1759062607 or create it here fore yourself: X Variable 1 37.2077457475 9.2959401573 4.0Dataset DATASET of Systolic Blood Pressure and weight measurements for 23 individuals Lecture #7 homework Subject # Body Weight Systolic BP (kg) (mmHg) Create your scatter plot somewhere here 1 74.4 109 2 85.1 114 3 78.3 94 4 77.2 109 5 63.8 104 6 60.9 98 7 82.2 116 8 99.8 121 9 78.0 111 10 71.8 116 11 90.2 115 12 105.4 133 13 100.4 128 14 80.9 128 15 81.8 105 16 109.0 127 17 90 120 18 69 110 19 96 136 20 67 118 21 99 132 22 74 111 23 73 112 Conclusions from scatter plot: (you can move this textbox by clicking on it) Step 5- Excel or Minitab Output Step 6 - Parameter Template Please enter the results from your regression analysis into this sheet. Please write your conclusions in the text boxes provided. R square R Standard Error of Residuals p value of residuals (regression analysis) Slope upper bound of confidence interval lower bound of confidence interval Confidence interval of slope (range: from to) Y intercept (no confidence interval needed) Equation of the regression line (format: y=bx+a) Increase of blood pressure in mmHg per kg of body weight Confidence interval of the increase (from to) Step 7 Fitted Line Plot Step 8 -Residuals analysis Copy the Residuals vs Fits Plot and the Normality Probability Plot here (no other plots please) Write your conclusions from these plots in the textbox provided) Conclusions about normality of residuals: Step 9 - Conclusions OVERALL CONCLUSIONS FROM REGRESSION ANALYSISClass 8 Homework Answer the question: Is there a linear relationship between body weight and blood pressure and if yes, by how much does the blood pressure increase for every 1 kg of body weight? Background: In the last homework you have calculated the association of obesity with hypertension, using the chi- square test for categorical variables. Obesity and hypertension status were defined by cutoff values and were just scored as yes and no. It is possible that there is a threshold or cutoff for weight above which blood pressure increases dramatically. However, it is more likely that blood pressure increases gradually with weight. And once the weight reaches obesity status the blood pressure often reaches hypertension levels (I assume you all know that hypertension means that the blood pressure is abnormally high). To test the hypothesis that blood pressure increases with weight, even before obesity and hypertension levels are reached, a pilot study was conducted obtaining the systolic blood pressure measurements and weights of 23 healthy individuals. (Note that the study is fictitious, real studies would have a lot more subjects.) Ensuring various assumptions are met for the statistical tests: The individuals were selected randomly, were not related, and data were collected in the same manner from each of the individuals. The study also controlled for other factors (besides weight) that may influence blood pressure (some of them with a known strong influence): · The individuals were all in their 20s and 30s (the average blood pressure increases slightly with age, more pronounced after the age of 60). · They did not have a family history of high blood pressure. · They were not know to be diabetic or suffer from kidney disease. Blood pressure measurements can fluctuate when taken several times in a row and may also depend on who is taking the blood pressure with which method. For this study (as is common for clinical studies), blood pressure was measured 3 times in a row over the period of ~ 1 hr (not just with the one time measurement that you get at the doctor’s office). So we make the assumption here that the measurements are reliable. Data set file (Excel): DATASET of Systolic Blood Pressure and weight measurements for 23 individuals.xls Note: In the data set file I created 5 tabs to enter your answers. The tabs are numbered according to the instructions below (Step 5: Excel or Minitab Output, Step 6: Parameter template, Step 7: Fitted Line Plot, Step 8: Residuals analysis, Step 9: Conclusions) Detailed instructions: 1. Rename the Excel file containing the dataset by adding your name. 2. Next to the dataset create a scatter plot (x axis: body weight, y axis: systolic blood pressure) for the data set. Give the plot a title and label your axis with the variable and the measurement unit (the latter in parenthesis). 3. From the scatter plot, determine visually if there seems to be a relationship between body wei