<p> Two Examples of ANOVA</p><p>Example 1: This is the age (response variable) by diagnosis (factor or explanatory variable) that was discussed in lecture on 5 Nov. I recoded the diagnosis codes as 0=Normal, 1=ACIS, and 2=AdCa. I entered the data into Minitab and performed the 1- way ANOVA (Stat → ANOVA → 1-way). I pressed the Graphs button and selected Boxplots of Data, Histograms of Residuals, Normal Plot of Residuals, and Residuals vs. Fits. In this case, Residuals vs. Order is meaningless since I don’t have the original order the observations were taken in. The results of the ANOVA are</p><p>One-way Analysis of Variance</p><p>Analysis of Variance for Age Source DF SS MS F P Diag 2 753 377 2.91 0.061 Error 67 8675 129 Total 69 9428 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev --+------+------+------+---- 0 13 46.08 15.28 (------*------) 1 38 37.84 8.99 (------*------) 2 19 42.58 12.66 (------*------) --+------+------+------+---- Pooled StDev = 11.38 35.0 40.0 45.0 50.0</p><p>Here are the Graphs that were produced: Normal Score</p><p>Diag Age -1.0 -0.5 -2.5 -2.0 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 20 30 40 50 60 70 80 -20 Normal Probability Plot of the Residuals the of Plot Probability Normal </p><p>0 -10 (means are indicated by solid circles) solid by indicated are (means Boxplots of Age by Diag by Age of Boxplots (response is Age) is (response 0 Residual</p><p>1 10 20</p><p>30 2 40 Histogram of the Residuals (response is Age)</p><p>20 y c n e</p><p> u 10 q e r F</p><p>0</p><p>-30 -20 -10 0 10 20 30 40 Residual</p><p>Residuals Versus the Fitted Values (response is Age)</p><p>40</p><p>30</p><p>20 l a</p><p> u 10 d i s e</p><p>R 0</p><p>-10</p><p>-20</p><p>37.5 38.5 39.5 40.5 41.5 42.5 43.5 44.5 45.5 46.5 Fitted Value</p><p>The Boxplots suggest the variances may not be equal. Computing the variances by the different diagnostic categories (Stat → Basic Statistics → Display Descriptive Statistics, then select Age for Variables, check By Variable, and select Diag): Descriptive Statistics</p><p>Variable Diag N Mean Median TrMean StDev Age 0 13 46.08 46.00 46.18 15.28 1 38 37.84 36.00 37.12 8.99 2 19 42.58 39.00 41.47 12.66</p><p>Variable Diag SE Mean Minimum Maximum Q1 Q3 Age 0 4.24 23.00 68.00 30.50 58.00 1 1.46 25.00 62.00 31.75 40.50 2 2.90 26.00 78.00 34.00 50.00</p><p>The ratio of the largest to smallest sample variance is (15.28/8.99)2 ≈ 2.9, which is a little problematic. The histogram and normal probability plot suggest that data may depart some from normality, but this is less problematic than the unequal variances.</p><p>Example 2: In this data set, we have a sample of patients with spinal injuries. The response variable is total hospital charge. The factor is “Financial Code” group, meaning basically type of insurance or method for payment (1 = Commercial Insurance, 2 = HMO, 3 = Medicaid, 4 = Medicare, 5 = Non-Resource (uninsured or self-insured), and 6 = PPO). The numbers of observations within each class:</p><p>Summary Statistics for Discrete Variables</p><p> group Count 1 396 2 74 3 82 4 61 5 190 6 97 N= 900</p><p>A 1-way ANOVA on the raw data, with the plots:</p><p>One-way Analysis of Variance</p><p>Analysis of Variance for charge Source DF SS MS F P group 5 3.351E+11 6.702E+10 9.22 0.000 Error 894 6.500E+12 7.271E+09 Total 899 6.835E+12 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev -----+------+------+------+- 1 396 41218 58974 (--*--) 2 74 34875 108495 (------*-----) 3 82 107746 122656 (-----*-----) 4 61 48728 75372 (------*------) 5 190 44153 70314 (---*---) 6 97 58861 135911 (-----*----) -----+------+------+------+- Pooled StDev = 85270 30000 60000 90000 120000</p><p>I made the font smaller so that everything fit without line wraparound.</p><p>Boxplots of charge by group (means are indicated by solid circles)</p><p>1000000 e g r a h</p><p> c 500000</p><p>0</p><p> group 1 2 3 4 5 6 Histogram of the Residuals (response is charge)</p><p>400</p><p>300 y c n e</p><p> u 200 q e r F</p><p>100</p><p>0</p><p>0 500000 1000000 Residual</p><p>Normal Probability Plot of the Residuals (response is charge)</p><p>3</p><p>2 e</p><p> r 1 o c S</p><p> l 0 a m r</p><p> o -1 N</p><p>-2</p><p>-3</p><p>0 500000 1000000 Residual Residuals Versus the Fitted Values (response is charge)</p><p>1000000 l a u d i</p><p> s 500000 e R</p><p>0</p><p>40000 50000 60000 70000 80000 90000 100000 110000 Fitted Value</p><p>Residuals Versus the Order of the Data (response is charge)</p><p>1000000 l a u d i</p><p> s 500000 e R</p><p>0</p><p>100 200 300 400 500 600 700 800 900 Observation Order</p><p>I also ran the analysis, but using the logarithm of the hospital charge as the response variable. Here are the results:</p><p>One-way Analysis of Variance Analysis of Variance for logcharg Source DF SS MS F P group 5 17.047 3.409 10.84 0.000 Error 894 281.169 0.315 Total 899 298.216 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ----+------+------+------+-- 1 396 4.3025 0.5312 (-*-) 2 74 4.0397 0.5969 (-----*----) 3 82 4.6809 0.6443 (----*----) 4 61 4.2820 0.6095 (----*-----) 5 190 4.3233 0.5328 (--*--) 6 97 4.3833 0.5962 (---*----) ----+------+------+------+-- Pooled StDev = 0.5608 4.00 4.25 4.50 4.75</p><p>Here are the Plots: Frequency group logcharge 20 30 40 50 60 70 10 3 4 5 6 0</p><p>1 -1 Boxplots of logcharg by group by logcharg of Boxplots Histogram of the Residuals the of Histogram (means are indicated by solid circles) solid by indicated are (means 2 (response is logcharg) is (response</p><p>0 3 Residual</p><p>4 1</p><p>5</p><p>6 2 Normal Probability Plot of the Residuals (response is logcharg)</p><p>3</p><p>2 e r 1 o c S l 0 a m r o -1 N</p><p>-2</p><p>-3</p><p>-1 0 1 2 Residual</p><p>Residuals Versus the Fitted Values (response is logcharg)</p><p>2</p><p>1 l a u d i s e 0 R</p><p>-1</p><p>4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Fitted Value Residuals Versus the Order of the Data (response is logcharg)</p><p>2</p><p>1 l a u d i s e 0 R</p><p>-1</p><p>100 200 300 400 500 600 700 800 900 Observation Order</p>