Prices of digital cameras

Prices of digital cameras The August 2012 issue of Consumer Reports included a report on digital cameras. The magazine listed 60 cameras, all of which were recommended by them, divided into six categories of increasing sophistication: Subcompact ( for those who need a camera that fits in a purse or pocket ), Compact ( for those who want the basics at a low price or advanced features ), Superzoom ( for those who need an extremely versatile zoom lens ), Point-and Shoot ( for those who want a built-in lens but advanced features similar to those on an SLR ), SLR-like ( for those who want a smaller, lighter camera with interchangeable lenses ), and SLR ( for those who want advanced features and performance, with interchangeable lenses and through-the-lens viewfinder ). It would be expected that different quality cameras would cost different amounts of money, but it is not clear which categories would actually involve a noticeable step-up in price. Since all of these cameras were recommended by the magazine, it is reasonable to assume that inherent quality is not that different between the different cameras, but certainly features could be. The following output summarizes the prices for each type. Note that the data are quite unbalanced, with as many as 16 cameras in one category and only 2 in another: Descriptive Statistics: Price Variable N N* Mean SE Mean StDev Minimum Price 1 (Subcompact) 11 0 300.0 25.2 83.4 180.0 2 (Compact) 7 0 267.1 25.9 68.5 160.0 3 (Superzoom) 16 0 323.1 34.9 139.5 200.0 4 (Point-and-shoot) 2 0 450.00 0.000000 0.000000 450.00 5 (SLR-like) 10 0 822.0 91.7 289.9 380.0 6 (SLR) 14 0 925 111 415 500 Variable Q1 Median Q3 Maximum Price 1 (Subcompact) 230.0 260.0 380.0 430.0 2 (Compact) 180.0 300.0 320.0 330.0 3 (Superzoom) 221.3 285.0 395.0 750.0 4 (Point-and-shoot) * 450.00 * 450.00 5 (SLR-like) 637.5 750.0 1025.0 1400.0 6 (SLR) 650 725 1163 1850 The three lowest categories seem similar to each other, with average prices around $300; in fact, Consumer Reports puts all three of these categories under basic cameras, and a first glance suggests that good-quality basic cameras cost on average around the same amount. c 2017, Jeffrey S. Simonoff 1

Point-and-shoot cameras cost a little more on average, but with only two cameras in that category it is hard to be sure. SLR-type and SLR cameras are quite noticeably more expensive. Side-by-side boxplots show that not only are there differences in prices between categories, there are also differences in variability, with more expensive cameras having more variable prices than less expensive ones. The following one-way ANOVA shows that there is, indeed, a statistically significant difference in prices between categories of cameras. General Linear Model: Price versus Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Fixed 6 1 (Subcompact), 2 (Compact), 3 (Superzoom), 4 (Point-and-shoot), 5 (SLR-like), 6 (SLR) Analysis of Variance c 2017, Jeffrey S. Simonoff 2

Source DF Adj SS Adj MS F-Value P-Value 5 4793902 958780 15.27 0.000 Error 54 3389547 62769 Total 59 8183448 Model Summary S R-sq R-sq(adj) R-sq(pred) 250.538 58.58% 54.75% 51.24% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 514.5 41.1 12.53 0.000 1 (Subcompact) -214.5 74.1-2.90 0.005 2.17 2 (Compact) -247.4 87.6-2.83 0.007 2.46 3 (Superzoom) -191.4 65.6-2.92 0.005 2.05 4 (Point-and-shoot) -65 150-0.43 0.669 4.90 5 (SLR-like) 307.5 76.6 4.01 0.000 2.22 Regression Equation Price = 514.5-214.5_1 (Subcompact) -247.4_2 (Compact) -191.4_3 (Superzoom) -65_4 (Point-and-shoot) +307.5_5 (SLR-like) +410.5_6 (SLR) Means Fitted Term Mean SE Mean 1 (Subcompact) 300.0 75.5 2 (Compact) 267.1 94.7 3 (Superzoom) 323.1 62.6 4 (Point-and-shoot) 450 177 5 (SLR-like) 822.0 79.2 6 (SLR) 925.0 67.0 Note that the VIFs refer to the underlying effect coding variables, and as such collinearity is not really a meaningful concept here. c 2017, Jeffrey S. Simonoff 3

Are all of the categories distinct? Definitely not: Comparisons for Price Tukey Pairwise Comparisons: Response = Price, Term = Grouping Information Using the Tukey Method and 95% Confidence N Mean Grouping 6 (SLR) 14 925.000 A 5 (SLR-like) 10 822.000 A 4 (Point-and-shoot) 2 450.000 A B 3 (Superzoom) 16 323.125 B 1 (Subcompact) 11 300.000 B 2 (Compact) 7 267.143 B Means that do not share a letter are significantly different. Tukey Simultaneous Tests for Differences of Means Difference SE of Simultaneous Difference of Levels of Means Difference 95% CI 2 (Compact) - 1 (Subcompact) -33 121 ( -391, 325) 3 (Superzoom) - 1 (Subcompact) 23.1 98.1 (-266.9, 313.2) 4 (Point-and-shoot) - 1 (Subcompact) 150 193 ( -419, 719) 5 (SLR-like) - 1 (Subcompact) 522 109 ( 198, 846) 6 (SLR) - 1 (Subcompact) 625 101 ( 327, 923) 3 (Superzoom) - 2 (Compact) 56 114 ( -280, 392) 4 (Point-and-shoot) - 2 (Compact) 183 201 ( -411, 777) 5 (SLR-like) - 2 (Compact) 555 123 ( 190, 920) 6 (SLR) - 2 (Compact) 658 116 ( 315, 1001) 4 (Point-and-shoot) - 3 (Superzoom) 127 188 ( -429, 682) 5 (SLR-like) - 3 (Superzoom) 499 101 ( 200, 797) 6 (SLR) - 3 (Superzoom) 601.9 91.7 ( 330.9, 872.9) 5 (SLR-like) - 4 (Point-and-shoot) 372 194 ( -202, 946) 6 (SLR) - 4 (Point-and-shoot) 475 189 ( -85, 1035) 6 (SLR) - 5 (SLR-like) 103 104 ( -204, 410) Adjusted Difference of Levels T-Value P-Value 2 (Compact) - 1 (Subcompact) -0.27 1.000 3 (Superzoom) - 1 (Subcompact) 0.24 1.000 4 (Point-and-shoot) - 1 (Subcompact) 0.78 0.970 c 2017, Jeffrey S. Simonoff 4

5 (SLR-like) - 1 (Subcompact) 4.77 0.000 6 (SLR) - 1 (Subcompact) 6.19 0.000 3 (Superzoom) - 2 (Compact) 0.49 0.996 4 (Point-and-shoot) - 2 (Compact) 0.91 0.942 5 (SLR-like) - 2 (Compact) 4.49 0.001 6 (SLR) - 2 (Compact) 5.67 0.000 4 (Point-and-shoot) - 3 (Superzoom) 0.68 0.984 5 (SLR-like) - 3 (Superzoom) 4.94 0.000 6 (SLR) - 3 (Superzoom) 6.56 0.000 5 (SLR-like) - 4 (Point-and-shoot) 1.92 0.403 6 (SLR) - 4 (Point-and-shoot) 2.51 0.140 6 (SLR) - 5 (SLR-like) 0.99 0.918 Individual confidence level = 99.54% As the descriptive statistics and boxplots suggested, the three categories of basic cameras are not at all distinct from each other with respect to price, but are from the SLR-type and SLR cameras. Before we take this too seriously, however, we need to go back and remember the apparent nonconstant variance form the boxplots, which a plot of residuals versus fitted values confirms: c 2017, Jeffrey S. Simonoff 5

The evidence for nonconstant variance is very clear from the plot (a long right tail also), but sometimes it isn t so obvious. What would be helpful would be a test for nonconstant variance, and it turns out that one is easily constructed using the residuals from the ANOVA fit. The test is called Levene s test (Minitab provides a closely-related version of Levene s test under Test for equal variances, but constructing it directly is more flexible, so you should do that, rather than using the built-in test). It is based on the simple idea that the variability of the errors corresponds to their spread, and a higher or lower average absolute value of the errors E( ε i ) should correspond to a higher or lower variance of the errors (which corresponds to E(ε 2 i )). Levene s test is thus constructed by saving the (standardized) residuals from the original ANOVA fit, taking their absolute values, and then using those absolute residuals as the response in an ANOVA. A statistically significant F -statistic corresponds to rejection of the hypothesis that the variability in the errors across levels of the grouping variable is constant. (The reason why the absolute residuals are used as the response rather than the squared errors is that the squared errors are more likely to be long right-tailed, making the test less trustworthy.) A Levene s test applied to these data confirms the nonconstant variance: General Linear Model: absres versus Method Factor coding (-1, 0, +1) c 2017, Jeffrey S. Simonoff 6

Factor Information Factor Type Levels Values Fixed 6 1 (Subcompact), 2 (Compact), 3 (Superzoom), 4 (Point-and-shoot), 5 (SLR-like), 6 (SLR) Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value 5 12.47 39.32% 12.47 2.4930 7.00 0.000 Error 54 19.23 60.68% 19.23 0.3562 Total 59 31.70 100.00% Model Summary S R-sq R-sq(adj) PRESS R-sq(pred) 0.596791 39.32% 33.71% 22.6397 28.58% Since variance increases with the level of the target variable, we might consider a transformation, such as taking logs. This looks much better (the two cameras in the Point-and-Shoot category had the same price, so there is no variability in observed priced for that group). Here are the results of an ANOVA: c 2017, Jeffrey S. Simonoff 7

General Linear Model: Logged price versus Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Fixed 6 1 (Subcompact), 2 (Compact), 3 (Superzoom), 4 (Point-and-shoot), 5 (SLR-like), 6 (SLR) Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value 5 2.928 0.58556 25.42 0.000 Error 54 1.244 0.02304 Total 59 4.172 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.151774 70.18% 67.42% 64.56% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 2.6379 0.0249 106.01 0.000 1 (Subcompact) -0.1762 0.0449-3.93 0.000 2.17 2 (Compact) -0.2259 0.0530-4.26 0.000 2.46 3 (Superzoom) -0.1580 0.0397-3.98 0.000 2.05 4 (Point-and-shoot) 0.0153 0.0911 0.17 0.867 4.90 5 (SLR-like) 0.2520 0.0464 5.43 0.000 2.22 Means Fitted Term Mean SE Mean c 2017, Jeffrey S. Simonoff 8

1 (Subcompact) 2.4617 0.0458 2 (Compact) 2.4121 0.0574 3 (Superzoom) 2.4799 0.0379 4 (Point-and-shoot) 2.653 0.107 5 (SLR-like) 2.8900 0.0480 6 (SLR) 2.9307 0.0406 The means are in the logged scale, so they need to be antilogged (which will produce geometric means for each group). So, for example, the typical price of a subcompact camera is 10 2.4617 = $290, while that of an SLR-like camera is 10 2.89 = $776. The differences between coefficients can be interpreted directly as estimated multiplicative differences between category prices. So, for example, an SLR camera is estimated to cost 2.82 times what a superzoom camera would cost (10 2.931 2.48 = 2.82). The same categories are viewed as statistically significantly different from each other in the logged scale: Comparisons for Logged price Tukey Pairwise Comparisons: Response = Logged price, Term = Grouping Information Using the Tukey Method and 95% Confidence N Mean Grouping 6 (SLR) 14 2.93069 A 5 (SLR-like) 10 2.88996 A 4 (Point-and-shoot) 2 2.65321 A B 3 (Superzoom) 16 2.47990 B 1 (Subcompact) 11 2.46171 B 2 (Compact) 7 2.41207 B Means that do not share a letter are significantly different. Tukey Simultaneous Tests for Differences of Means Difference SE of Simultaneous Difference of Levels of Means Difference 95% CI 2 (Compact) - 1 (Subcompact) -0.0496 0.0734 (-0.2665, 0.1672) 3 (Superzoom) - 1 (Subcompact) 0.0182 0.0594 (-0.1575, 0.1939) 4 (Point-and-shoot) - 1 (Subcompact) 0.192 0.117 ( -0.153, 0.536) 5 (SLR-like) - 1 (Subcompact) 0.4283 0.0663 ( 0.2322, 0.6243) c 2017, Jeffrey S. Simonoff 9

6 (SLR) - 1 (Subcompact) 0.4690 0.0612 ( 0.2882, 0.6497) 3 (Superzoom) - 2 (Compact) 0.0678 0.0688 (-0.1355, 0.2711) 4 (Point-and-shoot) - 2 (Compact) 0.241 0.122 ( -0.119, 0.601) 5 (SLR-like) - 2 (Compact) 0.4779 0.0748 ( 0.2568, 0.6990) 6 (SLR) - 2 (Compact) 0.5186 0.0703 ( 0.3110, 0.7263) 4 (Point-and-shoot) - 3 (Superzoom) 0.173 0.114 ( -0.163, 0.510) 5 (SLR-like) - 3 (Superzoom) 0.4101 0.0612 ( 0.2292, 0.5909) 6 (SLR) - 3 (Superzoom) 0.4508 0.0555 ( 0.2866, 0.6150) 5 (SLR-like) - 4 (Point-and-shoot) 0.237 0.118 ( -0.111, 0.584) 6 (SLR) - 4 (Point-and-shoot) 0.277 0.115 ( -0.062, 0.617) 6 (SLR) - 5 (SLR-like) 0.0407 0.0628 (-0.1450, 0.2265) Adjusted Difference of Levels T-Value P-Value 2 (Compact) - 1 (Subcompact) -0.68 0.984 3 (Superzoom) - 1 (Subcompact) 0.31 1.000 4 (Point-and-shoot) - 1 (Subcompact) 1.64 0.576 5 (SLR-like) - 1 (Subcompact) 6.46 0.000 6 (SLR) - 1 (Subcompact) 7.67 0.000 3 (Superzoom) - 2 (Compact) 0.99 0.920 4 (Point-and-shoot) - 2 (Compact) 1.98 0.366 5 (SLR-like) - 2 (Compact) 6.39 0.000 6 (SLR) - 2 (Compact) 7.38 0.000 4 (Point-and-shoot) - 3 (Superzoom) 1.52 0.652 5 (SLR-like) - 3 (Superzoom) 6.70 0.000 6 (SLR) - 3 (Superzoom) 8.12 0.000 5 (SLR-like) - 4 (Point-and-shoot) 2.01 0.348 6 (SLR) - 4 (Point-and-shoot) 2.42 0.168 6 (SLR) - 5 (SLR-like) 0.65 0.987 Individual confidence level = 99.54% c 2017, Jeffrey S. Simonoff 10

Residual plots exhibit a bit of a long right tail, but nonconstant variance seems to have been addressed: c 2017, Jeffrey S. Simonoff 11

General Linear Model: absres versus Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Fixed 6 1 (Subcompact), 2 (Compact), 3 (Superzoom), 4 (Point-and-shoot), 5 (SLR-like), 6 (SLR) Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value 5 2.037 10.67% 2.037 0.4075 1.29 0.282 Error 54 17.056 89.33% 17.056 0.3159 Total 59 19.093 100.00% Model Summary S R-sq R-sq(adj) PRESS R-sq(pred) 0.562006 10.67% 2.40% 20.2497 0.00% Diagnostics are given below. There are a couple of unusually expensive cameras (the Leica V-Lux 30, a $750 superzoom camera which is also the lowest-quality camera of that category that is recommended by Consumer Reports, and the Canon EOS 7D Digital, a $1850 SLR camera, by far the most expensive recommended camera). No points show up as influential according to Cook s distance, but the two point-and-shoot cameras are leverage points, with leverage values of 0.5 (remember that the cutoff value for the leverage is based on p = 5, since the ANOVA model used K 1 coding variables, with K = 6 here). This is because there are only two cameras in the group, so there isn t anything to do about it. Row Brand Model SRES2 HI2 COOK2 1 Nikon Coolpix S100-0.32298 0.090909 0.0017386 2 Canon PowerShot Elph 310 HS -0.32298 0.090909 0.0017386 c 2017, Jeffrey S. Simonoff 12

3 Canon PowerShot S95 0.81591 0.090909 0.0110952 4 Nikon Coolpix AW100 0.10648 0.090909 0.0001890 5 Canon PowerShot Elph 110 HS -0.69092 0.090909 0.0079562 6 Canon PowerShot SD4000 IS Elph -0.69092 0.090909 0.0079562 7 Panasonic Lumix DMC-3D1 0.96985 0.090909 0.0156768 8 Nikon Coolpix S8000-0.44068 0.090909 0.0032367 9 Canon PowerShot S100 0.81591 0.090909 0.0110952 10 Canon PowerShot Elph 300 HS -1.42656 0.090909 0.0339180 11 Nikon Coolpix S1200pj 1.18689 0.090909 0.0234785 12 Nikon Coolpix P300 0.75756 0.142857 0.0159414 13 Nikon Coolpix P310 0.46298 0.142857 0.0059542 14 Canon PowerShot Elph 510 HS 0.46298 0.142857 0.0059542 15 Nikon Coolpix S8200 0.24974 0.142857 0.0017325 16 Nikon Coolpix S8100-1.11584 0.142857 0.0345859 17 Panasonic Lumix DMC-TS4 0.66245 0.142857 0.0121900 18 Canon PowerShot SX130 IS -1.47987 0.142857 0.0608338 19 Panasonic Lumix DMC-FZ47-0.01891 0.062500 0.0000040 20 Nikon Coolpix P100 0.97546 0.062500 0.0105724 21 Nikon Coolpix S9100-1.21718 0.062500 0.0164615 22 Nikon Coolpix L120-0.22281 0.062500 0.0005516 23 Fujifilm FinePix F550EXR -0.80414 0.062500 0.0071850 24 Panasonic Lumix DMC-ZS20-0.11910 0.062500 0.0001576 25 Sony Cyber-shot DSC-HX9V 0.43665 0.062500 0.0021184 26 Canon PowerShot SX30 IS 1.11294 0.062500 0.0137626 27 Nikon Coolpix L110-0.93551 0.062500 0.0097243 28 Panasonic Lumix DMC-ZS15-0.86910 0.062500 0.0083926 29 Canon PowerShot SX40 HS 0.67968 0.062500 0.0051330 30 Olympus SZ-30MR -0.38553 0.062500 0.0016514 31 Panasonic Lumix DMC-ZS8-1.21718 0.062500 0.0164615 32 Nikon Coolpix P500 0.83127 0.062500 0.0076779 33 Panasonic Lumix DMC-ZS10-0.93551 0.062500 0.0097243 34 Leica V-Lux 30 2.68899 0.062500 0.0803407 35 Canon PowerShot G12-0.00000 0.500000 0.0000000 36 Panasonic Lumix DMC-FZ150-0.00000 0.500000 0.0000000 37 Sony SLT-A55VL -0.35501 0.100000 0.0023339 38 Sony SLT-A33L -0.77656 0.100000 0.0111675 39 Panasonic Lumix DMC-G2K -0.31161 0.100000 0.0017981 40 Nikon 1 V1 0.09116 0.100000 0.0001539 41 Sony SLT-A65VK 0.76421 0.100000 0.0108151 42 Sony NEX-7K 1.05169 0.100000 0.0204823 43 Sony SLT-A77V 1.77909 0.100000 0.0586138 44 Panasonic Lumix DMC-G3K -0.53513 0.100000 0.0053031 45 Samsung NX200 0.44642 0.100000 0.0036905 46 Panasonic Lumix DMC-GF2K -2.15425 0.100000 0.0859405 47 Canon EOS 600 0.61882 0.071429 0.0049094 c 2017, Jeffrey S. Simonoff 13

48 Olympus E-5 1.86959 0.071429 0.0448126 49 Nikon D7000 0.75695 0.071429 0.0073459 50 Canon EOS Rebel T3i -0.38033 0.071429 0.0018545 51 Canon EOS 7D Digital 2.30071 0.071429 0.0678622 52 Sony DSLR-A580L 0.16107 0.071429 0.0003326 53 Canon EOS Rebel T3-1.58434 0.071429 0.0321812 54 Canon EOS Rebel T2i -0.80526 0.071429 0.0083134 55 Nikon D3100-1.30132 0.071429 0.0217107 56 Pentax K-5 1.36508 0.071429 0.0238905 57 Nikon D5100-0.58520 0.071429 0.0043905 58 Pentax K-r -0.80526 0.071429 0.0083134 59 Sony DSLR-A560L -0.80526 0.071429 0.0083134 60 Nikon D5000-0.80526 0.071429 0.0083134 Consumer Reports occasionally labels a product a CR Best Buy if it thinks that the product is a particularly good deal for the consumer. Several of the cameras in the sample were so labeled, and unsurprisingly, most correspond to cameras with low (more negative residuals), as part of what makes something a good deal is that it is relatively inexpensive while still being good quality. Prediction intervals for each of the categories are as follows: Prediction for Logged price General Linear Model Information c 2017, Jeffrey S. Simonoff 14

Terms Variable Setting 1 (Subcompact) Fit SE Fit 95% CI 95% PI 2.46171 0.0457617 (2.36997, 2.55346) (2.14389, 2.77953) Variable Setting 2 (Compact) Fit SE Fit 95% CI 95% PI 2.41207 0.0573653 (2.29705, 2.52708) (2.08677, 2.73736) Variable Setting 3 (Superzoom) Fit SE Fit 95% CI 95% PI 2.47990 0.0379436 (2.40383, 2.55597) (2.16625, 2.79356) Variable Setting 4 (Point-and-shoot) Fit SE Fit 95% CI 95% PI 2.65321 0.107321 (2.43805, 2.86838) (2.28054, 3.02589) X c 2017, Jeffrey S. Simonoff 15

Variable Setting 5 (SLR-like) Fit SE Fit 95% CI 95% PI 2.88996 0.0479953 (2.79374, 2.98619) (2.57082, 3.20911) Variable Setting 6 (SLR) Fit SE Fit 95% CI 95% PI 2.93069 0.0405634 (2.84936, 3.01201) (2.61572, 3.24565) X denotes an unusual point relative to predictor levels used to fit the model. Recall that the data are in the logged scale, so the given intervals need to be antilogged. For a superzoom camera, for example, that gives ($146.64, $621.67). We could quit here, but a different approach would have been to stick with prices, and to try to address the observed nonconstant variance using weighted least squares. Using the standard deviations of the residuals separated by category, we create a weight variable corresponding to the inverse of the variance. The two point-and-shoot cameras with zero standard deviation can be given weights of 1; since their residuals are 0 they will not affect the fitting anyway. Descriptive Statistics: SRES1 Variable N N* Mean SE Mean StDev Minimum SRES1 1 (Subcompact) 11 0-0.000 0.105 0.349-0.502 2 (Compact) 7 0 0.000 0.112 0.295-0.462 3 (Superzoom) 16 0 0.000 0.144 0.575-0.508 4 (Point-and-shoot) 2 0 0.000000 0.000000 0.000000 0.000000 5 (SLR-like) 10 0-0.000 0.386 1.220-1.860 6 (SLR) 14 0 0.000 0.460 1.721-1.760 c 2017, Jeffrey S. Simonoff 16

Variable Q1 Median Q3 Maximum SRES1 1 (Subcompact) -0.293-0.167 0.335 0.544 2 (Compact) -0.376 0.142 0.228 0.271 3 (Superzoom) -0.420-0.157 0.296 1.760 4 (Point-and-shoot) * 0.000000 * 0.000000 5 (SLR-like) -0.776-0.303 0.854 2.432 6 (SLR) -1.139-0.828 0.984 3.831 Here are the results of a weighted one-way ANOVA: General Linear Model: Price versus Method Factor coding (-1, 0, +1) Weights wt Factor Information Factor Type Levels Values Fixed 6 1 (Subcompact), 2 (Compact), 3 (Superzoom), 4 (Point-and-shoot), 5 (SLR-like), 6 (SLR) Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value 5 3719130 743826 13.19 0.000 Error 54 3044311 56376 Total 59 6763441 Model Summary S R-sq R-sq(adj) R-sq(pred) 237.437 54.99% 50.82% 46.14% Coefficients Term Coef SE Coef T-Value P-Value VIF c 2017, Jeffrey S. Simonoff 17

Constant 514.5 37.6 13.67 0.000 1 (Subcompact) -214.5 42.8-5.01 0.000 2.07 2 (Compact) -247.4 43.4-5.70 0.000 2.02 3 (Superzoom) -191.4 46.8-4.09 0.000 1.75 4 (Point-and-shoot) -65 142-0.45 0.652 2.40 5 (SLR-like) 307.5 83.7 3.67 0.001 1.42 Regression Equation Price = 514.5-214.5_1 (Subcompact) -247.4_2 (Compact) -191.4_3 (Superzoom) -65_4 (Point-and-shoot) +307.5_5 (SLR-like) +410.5_6 (SLR) Means Fitted Term Mean SE Mean 1 (Subcompact) 300.0 25.0 2 (Compact) 267.1 26.5 3 (Superzoom) 323.1 34.1 4 (Point-and-shoot) 450 168 5 (SLR-like) 822.0 91.6 6 (SLR) 925 109 Tukey Pairwise Comparisons: Response = Price, Term = Grouping Information Using the Tukey Method and 95% Confidence N Mean Grouping 6 (SLR) 14 925.000 A 5 (SLR-like) 10 822.000 A 4 (Point-and-shoot) 2 450.000 A B 3 (Superzoom) 16 323.125 B 1 (Subcompact) 11 300.000 B 2 (Compact) 7 267.143 B Means that do not share a letter are significantly different. Tukey Simultaneous Tests for Differences of Means c 2017, Jeffrey S. Simonoff 18

Difference SE of Simultaneous Difference of Levels of Means Difference 95% CI 2 (Compact) - 1 (Subcompact) -32.9 36.4 (-140.5, 74.7) 3 (Superzoom) - 1 (Subcompact) 23.1 42.3 (-101.9, 148.1) 4 (Point-and-shoot) - 1 (Subcompact) 150 170 ( -352, 652) 5 (SLR-like) - 1 (Subcompact) 522.0 94.9 ( 241.4, 802.6) 6 (SLR) - 1 (Subcompact) 625 112 ( 294, 956) 3 (Superzoom) - 2 (Compact) 56.0 43.2 ( -71.7, 183.6) 4 (Point-and-shoot) - 2 (Compact) 183 170 ( -320, 685) 5 (SLR-like) - 2 (Compact) 554.9 95.3 ( 273.1, 836.7) 6 (SLR) - 2 (Compact) 658 112 ( 326, 990) 4 (Point-and-shoot) - 3 (Superzoom) 127 171 ( -380, 633) 5 (SLR-like) - 3 (Superzoom) 498.9 97.7 ( 210.0, 787.8) 6 (SLR) - 3 (Superzoom) 602 114 ( 264, 940) 5 (SLR-like) - 4 (Point-and-shoot) 372 191 ( -193, 937) 6 (SLR) - 4 (Point-and-shoot) 475 200 ( -117, 1067) 6 (SLR) - 5 (SLR-like) 103 142 ( -318, 524) Adjusted Difference of Levels T-Value P-Value 2 (Compact) - 1 (Subcompact) -0.90 0.944 3 (Superzoom) - 1 (Subcompact) 0.55 0.994 4 (Point-and-shoot) - 1 (Subcompact) 0.88 0.949 5 (SLR-like) - 1 (Subcompact) 5.50 0.000 6 (SLR) - 1 (Subcompact) 5.58 0.000 3 (Superzoom) - 2 (Compact) 1.30 0.786 4 (Point-and-shoot) - 2 (Compact) 1.08 0.889 5 (SLR-like) - 2 (Compact) 5.82 0.000 6 (SLR) - 2 (Compact) 5.86 0.000 4 (Point-and-shoot) - 3 (Superzoom) 0.74 0.976 5 (SLR-like) - 3 (Superzoom) 5.10 0.000 6 (SLR) - 3 (Superzoom) 5.26 0.000 5 (SLR-like) - 4 (Point-and-shoot) 1.95 0.387 6 (SLR) - 4 (Point-and-shoot) 2.37 0.185 6 (SLR) - 5 (SLR-like) 0.72 0.978 Individual confidence level = 99.54% c 2017, Jeffrey S. Simonoff 19

Once again, there is a highly statistically significant difference in prices for the different camera categories. Here are prediction intervals for the different groups; note that the weights need to be included, since the width of the interval is inversely proportional to the square of the weight. Prediction for Price General Linear Model Information Terms Variable Setting 1 (Subcompact) Fit SE Fit 95% CI 95% PI 300 24.9850 (249.908, 350.092) (126.476, 473.524) Weight = 8.21 c 2017, Jeffrey S. Simonoff 20

Variable Setting 2 (Compact) Fit SE Fit 95% CI 95% PI 267.143 26.4740 (214.066, 320.220) (117.018, 417.268) Weight = 11.491 Variable Setting 3 (Superzoom) Fit SE Fit 95% CI 95% PI 323.125 34.1291 (254.700, 391.550) (41.0025, 605.247) Weight = 3.025 Variable Setting 4 (Point-and-shoot) Fit SE Fit 95% CI 95% PI 450 167.893 (113.395, 786.605) (-133.017, 1033.02) X Weight = 1 Variable Setting 5 (SLR-like) Fit SE Fit 95% CI 95% PI 822 91.5932 (638.367, 1005.63) (212.958, 1431.04) c 2017, Jeffrey S. Simonoff 21

Weight = 0.672 Variable Setting 6 (SLR) Fit SE Fit 95% CI 95% PI 925 109.150 (706.167, 1143.83) (77.4621, 1772.54) Weight = 0.338 Note the prediction intervals for the more expensive cameras are much wider than the ones for the less expensive cameras, as should be the case (since the former cameras have higher variability in prices). The intervals still look a bit strange, however; for example, since the cheapest superzoom camera in the sample is only $200, this does not seem to be a very trustworthy interval (the interval based on logged prices seems much more reasonable), probably because the residuals are still right-tailed, although the nonconstant variance does seem to have been addressed very well: General Linear Model: absres versus c 2017, Jeffrey S. Simonoff 22

Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Fixed 6 1 (Subcompact), 2 (Compact), 3 (Superzoom), 4 (Point-and-shoot), 5 (SLR-like), 6 (SLR) Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value 5 1.463 0.2926 0.82 0.543 Error 54 19.352 0.3584 Total 59 20.814 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.598635 7.03% 0.00% 0.00% Note that both residual plots and the Levene s test must be based on standardized residuals, as those take the weights into account One final thing we might consider is that the rating classes are ordered here would a simple straight line relationship between logged price and category type be adequate? This corresponds to thinking that moving from say the first category (Subcompact) to the second (Compact) is the same (in terms of its relationship with prices) as moving from say the third (Superzoom) to the fourth (Point-and shoot). To be honest, I can t see any reason to think that, so under normal circumstances I would not do this last analysis. For pedagogical completeness I present it, but you shouldn t take it seriously. It is only sensible to do this type of analysis of there are contextual reasons to think of the categories this way. We can construct a partial F -test to compare the ANOVA fit to the linear relationship fit. First, here s the regression using an equally-spaced index variable defining the groups: Regression Analysis: Logged price versus number c 2017, Jeffrey S. Simonoff 23

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 2.5434 2.54340 90.60 0.000 number 1 2.5434 2.54340 90.60 0.000 Error 58 1.6283 0.02807 Lack-of-Fit 4 0.3844 0.09610 4.17 0.005 Pure Error 54 1.2439 0.02304 Total 59 4.1717 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.167553 60.97% 60.29% 58.24% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 2.2424 0.0478 46.93 0.000 number 0.1132 0.0119 9.52 0.000 1.00 Regression Equation Logged price = 2.2424 +0.1132number The partial F -test equals F = (1.6283 1.24392)/4 0.02304 = 4.17 on (4, 54) degrees of freedom, which has tail probability.005. That is, the ANOVA model fits better than the linear trend model. Note that we didn t actually have to do this calculation ourselves, since it is already given as the Pure error lack-of-fit test. Minitab commands To perform a one-way ANOVA, click on Stat ANOVA General Linear Model Fit General Linear Model. Enter the target variable under Responses:,and the variable c 2017, Jeffrey S. Simonoff 24

that defines the groups under Factors:. By default Minitab fits the model using effect codings, but you can change this to using indicator variables by clicking on Coding and then the radio button next to (1, 0). Note that you can choose the level you wish to be the reference level. You can save diagnostics by clicking on Storage, and can get plots by clicking on Graphs. Means for the groups are obtained by clicking on Options and clicking the drop-down box next to Means to say All terms in the model. Predictions are made as a followup analysis by clicking on Stat ANOVA General Linear Model Predict. You can enter in the categories you wish to predict under the categorical variable name. Note that if the ANOVA is a weighted analysis, you need to enter the appropriate weight value for each category. You could fit this model as a regression rather than a General Linear Model, except that then you would not be able to perform multiple comparisons. Multiple comparisons are obtained as a followup analysis by clicking on Stat ANOVA General Linear Model Comparisons. Enter the appropriate response variable using the drop-down menu next to Response. Highlight the variable that defines the groups in the dialog box below, and click on the button C = Compare levels for this item. Click on Graphs and check the box next to Interval plot for differences of means. Click on Results and check the box next to Tests and confidence intervals. To obtain a weighted analysis, click on Options and then enter the name of the weighting variable under Weights:. To create the weight variable, you can copy and paste the appropriate weight for each level into the cells in the Minitab worksheet for the observations in that level. You can also create the weight variable in the Calculator as a sum of indicator variables multiplied by weight values. To create all K indicators for a categorical variable click on Calc Make Indicator Variables and enter the variable name under Indicator variables for:. The program will choose default names for the indicators, but you can change them if you wish (to shorten them, for example, to Subcompact, Compact, Superzoom, Point-and-shoot, SLR-like, and SLR for these data). The weight variable on page 17 can then be constructed in the Calculator by creating the variable wt as Subcompact/(.349.349) + Compact/(.295.295) + Superzoom/(.575.575)+ Point and shoot + SLR like /(1.22 1.22) + SLR/(1.721 1.721) (note that the weight is set to 1 for the point-and-shoot cameras, as discussed earlier). If for some reason you wished to create effect codings, it is easy to do so once effect codings are constructed. If the code for SLR cameras is the one being omitted, then the c 2017, Jeffrey S. Simonoff 25

effect coding for subcompact cameras, for example, is Subcompact SLR. c 2017, Jeffrey S. Simonoff 26