---
title: "BANOVA examples"
author: "Chen Dong"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{BANOVA examples}
%\VignetteEngine{knitr::rmarkdown}
\usepackage[utf8]{inputenc}
---
```{r setup, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
```
We illustrate the application of the **BANOVA** package in a study into the influence
of color on gist perception of advertising, which is the very rapid identification of ads during
brief exposures. Specifically, we analyze the effect of color on the perception of the gist of ads
when the advertising exposure is brief and blurred (Wedel and Pieters 2015). In the study,
116 subjects were randomly assigned to one condition of a 5 (**blur**: normal, low, medium,
high, very high) x 2 (**color**: full color, grayscale) between-subjects, x 2 (**image**: typical ads, atypical ads) within-subjects, mixed design. Participants were exposed to 40 images, 32 fullpage
ads and 8 editorial pages. There were 8 ads per product category, with 4 typical and 4
atypical ones. Blur was manipulated by processing the advertising images with a Gaussian
blur filter of different radius. Subjects were flashed an image for 100msec. and then asked to
identify whether the image was an ad or not.
## Data
```{r, message=FALSE}
library(BANOVA)
data(colorad)
head(colorad)
```
The structure of colorad (data frame) is shown above using the `head()` function. It is in long format
including both within- subjects and between- subjects variables. Here, the within-subjects variable **typic** is a factor with 2 levels '0' (typical ads) and '1'(atypical ads); between-subjects variables are: **blur**, a numerical variable representing the blur of the image (the log-radius of a Gaussian blur filter used to produce the images); **blurfac**, a
factor variable with the five levels of blur; and **color**, a factor representing the color of the ads with 2 levels '0'(full color) and '1'(grayscale). id is the subject identification number. The dependent variable is the number of times ads were correctly identified as an ad, out of the 16 ads, for each subject for each level of typic.
## Models & results
We are interested in the effects of within- and between- subjects factors typic and color, and
the variable blur, as well as their interactions. The factor typic varies within individuals; the
factors blur, color and blur x color interaction vary between subjects.
The analysis of this experiment is executed with the function `BANOVA.run()` in the
**BANOVA** package (the continuous covariate blur is mean centered by default). The R code
to implement the analysis is shown below. For this model, the `BANOVA` function call needs
to include the binomial total (16) as an additional argument.
```{r, message=FALSE, eval=FALSE}
library(rstan)
set.seed(700)
model <- BANOVA.model(model_name = 'Binomial')
banova_fit <- BANOVA.build(model)
res <- BANOVA.run(y~typic, ~color*blur, data = colorad, fit = banova_fit,
id = 'id', num_trials = as.integer(16), iter = 1000, thin = 1, chains = 2)
```
The commands above load and build (compile) a Binomial Stan model included in the package. Then, the function `BANOVA.run` is called to run the simulation using two chains (1000 iterations each with thin = 1). Alternatively, users can set up the `model_name` argument to run the simulation directly using a precompiled model (note that it might report errors if compiler settings are not correct).
```{r, message=FALSE, eval=FALSE}
res_alt <- BANOVA.run(y~typic, ~color*blur, data = colorad, model_name = 'Binomial',
id = 'id', num_trials = as.integer(16), iter = 1000, thin = 1, chains = 2)
```
**ANOVA**-like table of sums of squares, effect sizes and p values, as well as the posterior means, standard
deviations, 95% credible intervals and p values of the parameters are produced with the
function `summary()`, the results are presented below. Note
that each of these tables now have two rows, one for each between-subject model (intercept
and typic).
```{r, message=FALSE, eval=FALSE}
summary(res)
#> Call:
#> BANOVA.run(l1_formula = y ~ typic, l2_formula = ~color * blur,
#> fit = banova_fit, data = colorad, id = 'id', iter = 1000,
#> num_trials = as.integer(16), thin = 1, chains = 2)
#>
#> Convergence diagnostics:
#> Geweke Diag. & Heidelberger and Welch's Diag.
#> Geweke stationarity test
#> (Intercept) : (Intercept) passed
#> (Intercept) : color1 passed
#> (Intercept) : blur passed
#> (Intercept) : color1:blur passed
#> typic1 : (Intercept) passed
#> typic1 : color1 passed
#> typic1 : blur passed
#> typic1 : color1:blur passed
#> Geweke convergence p value
#> (Intercept) : (Intercept) 0.1409
#> (Intercept) : color1 0.9288
#> (Intercept) : blur 0.4537
#> (Intercept) : color1:blur 0.9024
#> typic1 : (Intercept) 0.9642
#> typic1 : color1 0.9105
#> typic1 : blur 0.1451
#> typic1 : color1:blur 0.4603
#> H. & W. stationarity test
#> (Intercept) : (Intercept) passed
#> (Intercept) : color1 passed
#> (Intercept) : blur passed
#> (Intercept) : color1:blur passed
#> typic1 : (Intercept) passed
#> typic1 : color1 passed
#> typic1 : blur passed
#> typic1 : color1:blur passed
#> H. & W. convergence p value
#> (Intercept) : (Intercept) 0.4819
#> (Intercept) : color1 0.6041
#> (Intercept) : blur 0.7871
#> (Intercept) : color1:blur 0.4488
#> typic1 : (Intercept) 0.8005
#> typic1 : color1 0.8028
#> typic1 : blur 0.1004
#> typic1 : color1:blur 0.6634
#>
#> The Chain has converged.
#>
#> Table of sum of squares & effect sizes:
#>
#> Table of sum of squares:
#> (Intercept) color blur color:blur Residuals Total
#> (Intercept) 59.5532 2.3591 26.1587 9.0749 138.4249 229.8047
#> typic1 21.9510 1.9610 5.4139 7.0553 20.1769 50.6221
#>
#> Table of effect sizes (95% credible interval):
#> (Intercept) color
#> (Intercept) 0.3014 (0.249,0.358) 0.0159 (-0.015,0.104)
#> typic1 0.5215 (0.402,0.638) 0.0781 (-0.015,0.34)
#> blur color:blur
#> (Intercept) 0.1590 (0.108,0.21) 0.0581 (-0.006,0.199)
#> typic1 0.2114 (0.088,0.362) 0.2352 (0.006,0.521)
#>
#> Table of p-values (Multidimensional):
#> (Intercept) color blur color:blur
#> (Intercept) <0.0001 0.9560 <0.0001 0.1400
#> typic <0.0001 0.4180 <0.0001 0.0280
#>
#> Table of coefficients:
#> mean SD Quantile0.025 Quantile0.975
#> (Intercept) : (Intercept) 0.4975 0.0572 0.3881 0.6101
#> (Intercept) : color1 -0.0069 0.1245 -0.2605 0.2413
#> (Intercept) : blur -0.1716 0.0307 -0.2333 -0.1133
#> (Intercept) : color1:blur 0.0410 0.0291 -0.0171 0.1001
#> typic1 : (Intercept) 0.3011 0.0313 0.2374 0.3580
#> typic1 : color1 -0.0596 0.0736 -0.2062 0.0781
#> typic1 : blur -0.0767 0.0170 -0.1087 -0.0439
#> typic1 : color1:blur 0.0384 0.0178 0.0052 0.0733
#> p.value Signif.codes
#> (Intercept) : (Intercept) <0.0001 ***
#> (Intercept) : color1 0.9560
#> (Intercept) : blur <0.0001 ***
#> (Intercept) : color1:blur 0.1400
#> typic1 : (Intercept) <0.0001 ***
#> typic1 : color1 0.4180
#> typic1 : blur <0.0001 ***
#> typic1 : color1:blur 0.0280 *
#> ---
#> Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
#>
#> Multiple R-squared: 0.0664
#>
#> Table of predictions:
#>
#> Grand mean:
#> 9.95
#> 2.5% 97.5%
#> 9.5332 10.3673
#>
#>
#> typic mean 2.5% 97.5%
#> 0 11.035 10.5818 11.4579
#> 1 8.7829 8.2705 9.2933
#>
#>
#> color mean 2.5% 97.5%
#> 0 9.924 8.7817 10.9414
#> 1 9.9759 9.0196 10.8926
#>
#>
#> typic color mean 2.5% 97.5%
#> 0 0 10.8042 9.4812 11.9297
#> 0 1 11.26 10.202 12.271
#> 1 0 8.9913 7.7188 10.2174
#> 1 1 8.5735 7.4431 9.6343
```
Based on the above estimates, ad identification is significantly influenced by ad typicality
(typic): typical ads are identified more accurately as ads, compared to atypical ads. The
accuracy of ad identification is also affected by the degree of blur and its interaction with
typic. The three-factor interaction (blur x color x typic) is significant, which reveals that
color protects the identification of typical ads against blur, which is in line with the findings
of Wedel and Pieters (2015).
These results are based on the **BANOVA** model with blur as a continuous covariate. To
further understand the effects of blur, we can use the discrete variable blur (blurfac) in a
two-way **BANOVA** at the between-subjects level (and the factor typic again within-subjects),
using the following command:
```{r, message=FALSE, eval=FALSE}
library(rstan)
set.seed(900)
res_fac <- BANOVA.run(y~typic, ~color*blurfac, data = colorad, fit = banova_fit,
id = 'id', num_trials = as.integer(16), iter = 2000, thin = 1, chains = 2)
```
Since the above model involves more parameters, a larger number of iterations is used to ensure the chains for all parameters converge (iter = 2000).
```{r, message=FALSE, eval=FALSE}
summary(res_fac)
#> Call:
#> BANOVA.run(l1_formula = y ~ typic, l2_formula = ~color * blurfac,
#> fit = banova_fit, data = colorad, id = 'id', iter = 2000,
#> num_trials = as.integer(16), thin = 1, chains = 2)
#>
#> Convergence diagnostics:
#> Geweke Diag. & Heidelberger and Welch's Diag.
#> Geweke stationarity test
#> (Intercept) : (Intercept) passed
#> (Intercept) : color1 passed
#> (Intercept) : blurfac1 passed
#> (Intercept) : blurfac2 passed
#> (Intercept) : blurfac3 passed
#> (Intercept) : blurfac4 passed
#> (Intercept) : color1:blurfac1 passed
#> (Intercept) : color1:blurfac2 passed
#> (Intercept) : color1:blurfac3 passed
#> (Intercept) : color1:blurfac4 passed
#> typic1 : (Intercept) passed
#> typic1 : color1 passed
#> typic1 : blurfac1 passed
#> typic1 : blurfac2 passed
#> typic1 : blurfac3 passed
#> typic1 : blurfac4 passed
#> typic1 : color1:blurfac1 passed
#> typic1 : color1:blurfac2 passed
#> typic1 : color1:blurfac3 passed
#> typic1 : color1:blurfac4 passed
#> Geweke convergence p value
#> (Intercept) : (Intercept) 0.0217
#> (Intercept) : color1 0.4861
#> (Intercept) : blurfac1 0.7050
#> (Intercept) : blurfac2 0.5498
#> (Intercept) : blurfac3 0.4843
#> (Intercept) : blurfac4 0.6209
#> (Intercept) : color1:blurfac1 0.9840
#> (Intercept) : color1:blurfac2 0.6900
#> (Intercept) : color1:blurfac3 0.2931
#> (Intercept) : color1:blurfac4 0.7919
#> typic1 : (Intercept) 0.0159
#> typic1 : color1 0.5622
#> typic1 : blurfac1 0.0594
#> typic1 : blurfac2 0.1408
#> typic1 : blurfac3 0.0534
#> typic1 : blurfac4 0.3087
#> typic1 : color1:blurfac1 0.3151
#> typic1 : color1:blurfac2 0.4267
#> typic1 : color1:blurfac3 0.1776
#> typic1 : color1:blurfac4 0.8341
#> H. & W. stationarity test
#> (Intercept) : (Intercept) passed
#> (Intercept) : color1 passed
#> (Intercept) : blurfac1 passed
#> (Intercept) : blurfac2 passed
#> (Intercept) : blurfac3 passed
#> (Intercept) : blurfac4 passed
#> (Intercept) : color1:blurfac1 passed
#> (Intercept) : color1:blurfac2 passed
#> (Intercept) : color1:blurfac3 passed
#> (Intercept) : color1:blurfac4 passed
#> typic1 : (Intercept) passed
#> typic1 : color1 passed
#> typic1 : blurfac1 passed
#> typic1 : blurfac2 passed
#> typic1 : blurfac3 passed
#> typic1 : blurfac4 passed
#> typic1 : color1:blurfac1 passed
#> typic1 : color1:blurfac2 passed
#> typic1 : color1:blurfac3 passed
#> typic1 : color1:blurfac4 passed
#> H. & W. convergence p value
#> (Intercept) : (Intercept) 0.2829
#> (Intercept) : color1 0.8963
#> (Intercept) : blurfac1 0.5252
#> (Intercept) : blurfac2 0.8212
#> (Intercept) : blurfac3 0.5227
#> (Intercept) : blurfac4 0.9169
#> (Intercept) : color1:blurfac1 0.5844
#> (Intercept) : color1:blurfac2 0.4926
#> (Intercept) : color1:blurfac3 0.7559
#> (Intercept) : color1:blurfac4 0.8684
#> typic1 : (Intercept) 0.0515
#> typic1 : color1 0.0775
#> typic1 : blurfac1 0.2471
#> typic1 : blurfac2 0.1948
#> typic1 : blurfac3 0.0845
#> typic1 : blurfac4 0.0610
#> typic1 : color1:blurfac1 0.1983
#> typic1 : color1:blurfac2 0.2473
#> typic1 : color1:blurfac3 0.2181
#> typic1 : color1:blurfac4 0.8831
#>
#> The Chain has converged.
#>
#> Table of sum of squares & effect sizes:
#>
#> Table of sum of squares:
#> (Intercept) color blurfac color:blurfac Residuals Total
#> (Intercept) 59.3470 5.1145 29.5136 1.0276 136.9923 231.8971
#> typic1 21.3932 2.0202 13.6640 1.9275 16.7336 55.6143
#>
#> Table of effect sizes (95% credible interval):
#> (Intercept) color blurfac
#> (Intercept) 0.3028 (0.25,0.36) 0.0361 (0.01,0.066) 0.1774 (0.123,0.232)
#> typic1 0.5602 (0.444,0.67) 0.1075 (0.017,0.231) 0.4476 (0.315,0.585)
#> color:blurfac
#> (Intercept) 0.0075 (-0.028,0.038)
#> typic1 0.1023 (0.009,0.231)
#>
#> Table of p-values (Multidimensional):
#> (Intercept) color blurfac color:blurfac
#> (Intercept) <0.0001 0.0100 <0.0001 0.1590
#> typic <0.0001 0.0010 <0.0001 0.0200
#>
#> Table of coefficients:
#> mean SD Quantile0.025 Quantile0.975
#> (Intercept) : (Intercept) 0.4981 0.0574 0.3840 0.6115
#> (Intercept) : color1 0.1532 0.0567 0.0404 0.2683
#> (Intercept) : blurfac1 0.5591 0.1139 0.3470 0.7807
#> (Intercept) : blurfac2 0.0554 0.1168 -0.1731 0.2883
#> (Intercept) : blurfac3 0.0193 0.1036 -0.1775 0.2226
#> (Intercept) : blurfac4 -0.0565 0.1094 -0.2689 0.1672
#> (Intercept) : color1:blurfac1 -0.1586 0.1146 -0.3897 0.0724
#> (Intercept) : color1:blurfac2 0.1137 0.1162 -0.1094 0.3347
#> (Intercept) : color1:blurfac3 -0.0774 0.1096 -0.2961 0.1376
#> (Intercept) : color1:blurfac4 0.0125 0.1114 -0.2098 0.2329
#> typic1 : (Intercept) 0.3092 0.0318 0.2481 0.3742
#> typic1 : color1 0.0906 0.0311 0.0296 0.1532
#> typic1 : blurfac1 0.1621 0.0634 0.0379 0.2901
#> typic1 : blurfac2 0.3612 0.0692 0.2138 0.4951
#> typic1 : blurfac3 -0.0260 0.0642 -0.1543 0.0996
#> typic1 : blurfac4 -0.1653 0.0625 -0.2830 -0.0409
#> typic1 : color1:blurfac1 -0.1443 0.0645 -0.2727 -0.0231
#> typic1 : color1:blurfac2 0.0344 0.0671 -0.1041 0.1637
#> typic1 : color1:blurfac3 -0.0238 0.0597 -0.1456 0.0915
#> typic1 : color1:blurfac4 0.0956 0.0619 -0.0272 0.2211
#> p.value Signif.codes
#> (Intercept) : (Intercept) <0.0001 ***
#> (Intercept) : color1 0.0100 **
#> (Intercept) : blurfac1 <0.0001 ***
#> (Intercept) : blurfac2 0.6400
#> (Intercept) : blurfac3 0.8430
#> (Intercept) : blurfac4 0.5910
#> (Intercept) : color1:blurfac1 0.1590
#> (Intercept) : color1:blurfac2 0.3360
#> (Intercept) : color1:blurfac3 0.4850
#> (Intercept) : color1:blurfac4 0.8890
#> typic1 : (Intercept) <0.0001 ***
#> typic1 : color1 0.0010 ***
#> typic1 : blurfac1 0.0080 **
#> typic1 : blurfac2 <0.0001 ***
#> typic1 : blurfac3 0.6920
#> typic1 : blurfac4 0.0060 **
#> typic1 : color1:blurfac1 0.0200 *
#> typic1 : color1:blurfac2 0.5940
#> typic1 : color1:blurfac3 0.6940
#> typic1 : color1:blurfac4 0.1150
#> ---
#> Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
#>
#> Multiple R-squared: 0.0856
#>
#> Table of predictions:
#>
#> Grand mean:
#> 9.9522
#> 2.5% 97.5%
#> 9.5174 10.3726
#>
#>
#> typic mean 2.5% 97.5%
#> 0 11.0645 10.5896 11.4879
#> 1 8.7534 8.2242 9.2339
#>
#>
#> color mean 2.5% 97.5%
#> 0 10.5167 9.9177 11.0551
#> 1 9.3662 8.7386 9.9471
#>
#> blurfac mean 2.5% 97.5%
#> 1 11.8745 11.0664 12.6146
#> 2 10.1591 9.1897 11.1158
#> 3 10.0247 9.1808 10.8356
#> 4 9.7383 8.8491 10.6451
#> 5 7.6834 6.6721 8.65
#>
#>
#> typic color mean 2.5% 97.5%
#> 0 0 11.8558 11.2628 12.3795
#> 0 1 10.1961 9.521 10.8499
#> 1 0 9.0004 8.262 9.6717
#> 1 1 8.5049 7.8023 9.2008
#>
#> typic blurfac mean 2.5% 97.5%
#> 0 1 13.1487 12.3854 13.7718
#> 0 2 12.3638 11.4074 13.14
#> 0 3 11.0416 10.0703 11.8974
#> 0 4 10.2774 9.2503 11.2558
#> 0 5 7.5923 6.4556 8.7182
#> 1 1 10.2786 9.1893 11.2832
#> 1 2 7.533 6.3988 8.7
#> 1 3 8.9327 7.9134 9.9488
#> 1 4 9.1821 8.1045 10.2629
#> 1 5 7.7745 6.6116 8.9589
#>
#>
#> color blurfac mean 2.5% 97.5%
#> 0 1 11.8576 10.7518 12.8309
#> 1 1 11.8912 10.7507 12.8908
#> 0 2 11.1088 9.6862 12.2988
#> 1 2 9.1388 7.7334 10.5106
#> 0 3 10.3056 9.0203 11.4538
#> 1 3 9.7384 8.5515 10.9235
#> 0 4 10.3571 8.9993 11.6076
#> 1 4 9.0969 7.7976 10.4325
#> 0 5 8.7333 7.3643 10.1124
#> 1 5 6.6442 5.314 8.0042
#>
#>
#> typic color blurfac mean 2.5% 97.5%
#> 0 0 1 13.0075 11.8829 13.8698
#> 0 1 1 13.2845 12.2295 14.1165
#> 0 0 2 13.3472 12.179 14.2525
#> 0 1 2 11.1483 9.6802 12.4133
#> 0 0 3 11.5161 10.1419 12.6794
#> 0 1 3 10.5408 9.1534 11.8042
#> 0 0 4 11.4974 10.0861 12.6792
#> 0 1 4 8.9302 7.4175 10.3435
#> 0 0 5 9.1508 7.5622 10.651
#> 0 1 5 6.0642 4.63 7.6654
#> 1 0 1 10.4543 8.9691 11.7715
#> 1 1 1 10.1005 8.5915 11.5018
#> 1 0 2 8.0995 6.3761 9.7681
#> 1 1 2 6.9711 5.4889 8.5847
#> 1 0 3 8.9678 7.3996 10.3327
#> 1 1 3 8.8976 7.5123 10.3464
#> 1 0 4 9.1014 7.5952 10.681
#> 1 1 4 9.2626 7.7261 10.8305
#> 1 0 5 8.3118 6.8199 9.9193
#> 1 1 5 7.2393 5.6957 8.8424
```
We first inspect the tables of sums of squares and effect sizes. In these tables, the columns
denote between-subjects factors and the rows denote the within-subjects factors. The values
in the table present the sum-of-squares and effect sizes of the effects of these factors. Again,
the accuracy of ad identification is affected by blur, and to a lesser extent by color. From
the tables of p values, ad typicality (the value corresponding to the row name 'typic' and
column name '(Intercept)') and the degree of blur (the value corresponding to the row name
'(Intercept)' and column name 'blurfac') are again highly significant. There is also support
for the main effect of color. The three-factor interaction (blurfac x color x typic) is also
significant, which again shows that color protects the identification of typical ads against blur
(Wedel and Pieters 2015). The conclusions from the table of estimates are similar to those
from the results of the previous model, but this table for nonlinear effects of blur allows us
to inspect the effects of each level of bur, and the interactive effects with color and typicality.
Through the tables of predictions for all factors and their interactions, we can inspect these
effects in more detail. We can see that typical color ads (typic = 0, color = 0) are always
more accurately identified than atypical color ads (typic = 1, color = 0). Typical grayscale
ads (typic = 0, color = 1, blur = 1,...,5), however, are only more accurately identified than
atypical grayscale ads (typic = 1, color = 1, blur = 1,...,5) when there is no blur, or a low
level of blur (Wedel and Pieters 2015).
## References
- Wedel M and Pieters R (2015). "The Buffer Effect: The Role of Color When Advertising Exposures
Are Brief and Blurred." Marketing Science, 34(1), 134-143.
- Dong C and Wedel M (2017). "BANOVA: An R Package for Hierarchical Bayesian ANOVA."Journal of Statistical Software, 81(9), pp. 1-46.