Multivariate Analysis

Multivariate analysis refers to statistical techniques used to analyze data that involves multiple variables simultaneously
Rather than examining each variable in isolation, multivariate methods consider the relationships and interactions among all variables

Multivariate Analysis

Widely used in fields like marketing research, ecology, psychology, medicine, and finance where complex datasets with many variables are common
Helps researchers identify patterns, test hypotheses, and make predictions based on multiple variables simultaneously

Multivariate Regression

We have run regression with one independent variable (x) only
When the variance explained of a model is low, we assume that there are other factors that explain the response variable that are not being considered in the model
Multivariate Regression: Examines relationships between multiple independent variables and one or more dependent variables.

Data

We will be working with the same data on movies and their rating.

library(tidyverse)
all_media <- read_csv("data/movies.csv")

Question

What is the effect of type and genre on rating?

Inferential Statistics

model <- lm(averageRating ~ genre + type,
            data = all_media)
summary(model)


Call:
lm(formula = averageRating ~ genre + type, data = all_media)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.9196 -0.6299  0.1017  0.7512  4.9461 

Coefficients:
                  Estimate Std. Error  t value Pr(>|t|)    
(Intercept)       5.980290   0.004001 1494.848  < 2e-16 ***
genreAdult       -0.146381   0.010808  -13.544  < 2e-16 ***
genreAdventure    0.009168   0.005931    1.546 0.122113    
genreAnimation   -0.241429   0.011104  -21.742  < 2e-16 ***
genreBiography    0.628527   0.008904   70.589  < 2e-16 ***
genreComedy       0.039077   0.004068    9.607  < 2e-16 ***
genreCrime        0.118833   0.005486   21.661  < 2e-16 ***
genreDocumentary  0.640106   0.004993  128.189  < 2e-16 ***
genreDrama        0.268465   0.004416   60.793  < 2e-16 ***
genreFamily      -0.014477   0.010263   -1.411 0.158361    
genreFantasy      0.026625   0.014903    1.786 0.074021 .  
genreFilm-Noir    0.364710   0.269527    1.353 0.176009    
genreGame-Show   -0.054079   0.011768   -4.595 4.32e-06 ***
genreHistory      0.547596   0.035981   15.219  < 2e-16 ***
genreHorror      -0.850390   0.008641  -98.410  < 2e-16 ***
genreMusical      0.610299   0.012929   47.205  < 2e-16 ***
genreMystery      0.061452   0.019323    3.180 0.001472 ** 
genreNews        -1.343697   0.027049  -49.677  < 2e-16 ***
genreReality-TV  -0.243151   0.008315  -29.243  < 2e-16 ***
genreRomance      0.059879   0.016413    3.648 0.000264 ***
genreSci-Fi      -0.178261   0.017747  -10.045  < 2e-16 ***
genreShort       -0.080398   0.009259   -8.683  < 2e-16 ***
genreSport        0.204031   0.024078    8.474  < 2e-16 ***
genreTalk-Show   -0.501142   0.019687  -25.455  < 2e-16 ***
genreThriller    -0.188485   0.013455  -14.008  < 2e-16 ***
genreWar          0.080935   0.040708    1.988 0.046792 *  
genreWestern      0.197201   0.012857   15.338  < 2e-16 ***
typeshort         0.746905   0.004148  180.046  < 2e-16 ***
typetvEpisode     1.299175   0.003038  427.574  < 2e-16 ***
typetvMiniSeries  0.880923   0.012215   72.121  < 2e-16 ***
typetvMovie       0.305417   0.006107   50.015  < 2e-16 ***
typetvSeries      0.771852   0.005780  133.532  < 2e-16 ***
typetvShort       0.729547   0.025748   28.334  < 2e-16 ***
typetvSpecial     0.783455   0.014180   55.251  < 2e-16 ***
typevideo         0.417346   0.006803   61.346  < 2e-16 ***
typevideoGame     0.814507   0.104161    7.820 5.30e-15 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.205 on 1071553 degrees of freedom
Multiple R-squared:  0.1892,    Adjusted R-squared:  0.1892 
F-statistic:  7144 on 35 and 1071553 DF,  p-value: < 2.2e-16

Visualizing Results

What is the effect of type and genre on rating?

library(effects)

effect("type", model) |>
  data.frame() |>
  ggplot(aes(x = fit, xmin = lower, xmax = upper,
             y = reorder(type, fit), label = round(fit, 2))) +
  geom_errorbar() +
  geom_label()

Visualizing Results

What is the effect of type and genre on rating?

effect("genre", model) |>
  data.frame() |>
  ggplot(aes(x = fit, xmin = lower, xmax = upper,
             y = reorder(genre, fit))) +
  geom_errorbar() +
  geom_point()

Question

What factors have an effect on rating?

Inferential Statistics

model <- lm(averageRating ~ genre + type + year + runtime + numVotes,
            data = all_media)
summary(model)


Call:
lm(formula = averageRating ~ genre + type + year + runtime + 
    numVotes, data = all_media)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.9748 -0.6314  0.1068  0.7494  4.9290 

Coefficients:
                   Estimate Std. Error  t value Pr(>|t|)    
(Intercept)      -4.686e+00  1.112e-01  -42.132  < 2e-16 ***
genreAdult       -1.078e-01  1.075e-02  -10.021  < 2e-16 ***
genreAdventure    3.851e-02  5.904e-03    6.522 6.94e-11 ***
genreAnimation   -1.929e-01  1.105e-02  -17.455  < 2e-16 ***
genreBiography    5.944e-01  8.863e-03   67.066  < 2e-16 ***
genreComedy       5.881e-02  4.050e-03   14.521  < 2e-16 ***
genreCrime        1.309e-01  5.457e-03   23.989  < 2e-16 ***
genreDocumentary  6.187e-01  4.984e-03  124.144  < 2e-16 ***
genreDrama        2.760e-01  4.395e-03   62.803  < 2e-16 ***
genreFamily       3.520e-02  1.022e-02    3.445 0.000571 ***
genreFantasy      1.054e-02  1.482e-02    0.711 0.477009    
genreFilm-Noir    6.351e-01  2.680e-01    2.370 0.017812 *  
genreGame-Show   -9.689e-02  1.172e-02   -8.270  < 2e-16 ***
genreHistory      5.363e-01  3.578e-02   14.988  < 2e-16 ***
genreHorror      -9.003e-01  8.616e-03 -104.497  < 2e-16 ***
genreMusical      6.594e-01  1.286e-02   51.258  < 2e-16 ***
genreMystery      5.447e-02  1.922e-02    2.834 0.004593 ** 
genreNews        -1.330e+00  2.690e-02  -49.453  < 2e-16 ***
genreReality-TV  -2.832e-01  8.285e-03  -34.184  < 2e-16 ***
genreRomance      3.815e-02  1.633e-02    2.336 0.019471 *  
genreSci-Fi      -2.076e-01  1.765e-02  -11.757  < 2e-16 ***
genreShort       -6.313e-02  9.209e-03   -6.854 7.16e-12 ***
genreSport        2.078e-01  2.394e-02    8.676  < 2e-16 ***
genreTalk-Show   -5.269e-01  1.958e-02  -26.907  < 2e-16 ***
genreThriller    -2.383e-01  1.340e-02  -17.784  < 2e-16 ***
genreWar          1.506e-01  4.049e-02    3.720 0.000199 ***
genreWestern      4.418e-01  1.302e-02   33.931  < 2e-16 ***
typeshort         7.434e-01  4.133e-03  179.875  < 2e-16 ***
typetvEpisode     1.255e+00  3.089e-03  406.313  < 2e-16 ***
typetvMiniSeries  8.255e-01  1.216e-02   67.868  < 2e-16 ***
typetvMovie       2.981e-01  6.078e-03   49.049  < 2e-16 ***
typetvSeries      7.263e-01  5.772e-03  125.845  < 2e-16 ***
typetvShort       7.175e-01  2.561e-02   28.022  < 2e-16 ***
typetvSpecial     7.242e-01  1.412e-02   51.289  < 2e-16 ***
typevideo         3.763e-01  6.788e-03   55.444  < 2e-16 ***
typevideoGame     7.620e-01  1.036e-01    7.357 1.89e-13 ***
year              5.335e-03  5.570e-05   95.780  < 2e-16 ***
runtime           3.691e-07  3.245e-07    1.137 0.255373    
numVotes          2.798e-06  5.444e-08   51.392  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.199 on 1071550 degrees of freedom
Multiple R-squared:  0.1982,    Adjusted R-squared:  0.1982 
F-statistic:  6972 on 38 and 1071550 DF,  p-value: < 2.2e-16

Inferential Statistics

just_movies <- all_media |>
  filter(type == "movie")

model_movie <- lm(averageRating ~ genre + year + numVotes + runtime,
                  data = just_movies)
summary(model_movie)


Call:
lm(formula = averageRating ~ genre + year + numVotes + runtime, 
    data = just_movies)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.0407 -0.7211  0.0701  0.7729  5.2770 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)       5.913e+00  1.841e-01  32.112  < 2e-16 ***
genreAdult       -1.857e-01  2.109e-02  -8.802  < 2e-16 ***
genreAdventure    2.703e-01  1.326e-02  20.385  < 2e-16 ***
genreAnimation    5.166e-01  4.352e-02  11.871  < 2e-16 ***
genreBiography    1.279e+00  1.465e-02  87.289  < 2e-16 ***
genreComedy       2.175e-01  8.610e-03  25.257  < 2e-16 ***
genreCrime        3.645e-01  1.237e-02  29.459  < 2e-16 ***
genreDocumentary  1.548e+00  9.429e-03 164.195  < 2e-16 ***
genreDrama        5.690e-01  8.278e-03  68.736  < 2e-16 ***
genreFamily       3.598e-01  2.602e-02  13.827  < 2e-16 ***
genreFantasy     -7.852e-03  3.094e-02  -0.254 0.799679    
genreFilm-Noir    6.699e-01  2.729e-01   2.455 0.014098 *  
genreGame-Show    1.736e+00  1.220e+00   1.423 0.154711    
genreHistory      1.056e+00  7.174e-02  14.721  < 2e-16 ***
genreHorror      -9.357e-01  1.313e-02 -71.242  < 2e-16 ***
genreMusical      9.611e-01  3.010e-02  31.927  < 2e-16 ***
genreMystery      1.774e-01  3.240e-02   5.475 4.38e-08 ***
genreNews         1.492e+00  4.980e-01   2.995 0.002742 ** 
genreReality-TV   7.571e-01  2.393e-01   3.163 0.001560 ** 
genreRomance      3.628e-01  2.443e-02  14.853  < 2e-16 ***
genreSci-Fi      -4.333e-01  3.689e-02 -11.743  < 2e-16 ***
genreSport        9.082e-01  8.592e-02  10.570  < 2e-16 ***
genreTalk-Show    9.112e-01  4.611e-01   1.976 0.048154 *  
genreThriller    -7.378e-02  1.989e-02  -3.709 0.000208 ***
genreWar          2.902e-01  5.721e-02   5.073 3.93e-07 ***
genreWestern      1.486e-01  2.672e-02   5.563 2.66e-08 ***
year             -1.301e-04  9.217e-05  -1.411 0.158220    
numVotes          2.943e-06  5.848e-08  50.327  < 2e-16 ***
runtime           9.125e-05  1.312e-05   6.956 3.50e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.22 on 287315 degrees of freedom
Multiple R-squared:  0.1847,    Adjusted R-squared:  0.1846 
F-statistic:  2324 on 28 and 287315 DF,  p-value: < 2.2e-16

Inferential Statistics

anova(model_movie)

Case Study

Download data on American Idol, set up your analysis environment, and answer the following question:

What factors have an effect on episode rating?