Multivariate Analysis Report

Connor King 2023-05-17

Dataset
Project Goal
- Variable Descriptions
Exploratory Data Analysis
Multiple Multivariate Regression
One-Way MANOVA

Dataset

Description

The dataset is a combination of data from Spotify and Youtube and consists of several attributes.

The Spotify data includes various characteristics of songs, like Danceability, Energy, Key, Loudness, Speechiness, Acousticness, Instrumentalness, Liveness, Valence, Tempo, and Duration_ms.

The Youtube data, on the other hand, consists of dependent variables that measure the popularity of these songs on Youtube where the number of Views, Likes, Comments are tracked for the corresponding music video.

The dependent variable of Stream was from Spotify which represents the number of times a particular song or track has been played or listened to on Spotify.

## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ✔ forcats   1.0.0     ✔ stringr   1.5.0
## ✔ ggplot2   3.4.2     ✔ tibble    3.2.1
## ✔ lubridate 1.9.2     ✔ tidyr     1.3.0
## ✔ purrr     1.0.1     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
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## New names:
## Rows: 20718 Columns: 28
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (10): Artist, Url_spotify, Track, Album, Album_type, Uri, Url_youtube, T...
## dbl (16): ...1, Danceability, Energy, Key, Loudness, Speechiness, Acousticne...
## lgl  (2): Licensed, official_video
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

Project Goal

In this project I will seek to answer various questions via multivariate analysis. I will attempt to answer:

Can we predict the popularity of a song with the help of Spotify audio features?

I will achieve this via multiple multivariate regression analysis and include a bootstrap study to calculate confidence intervals for the R-squared values.

Are there differences in the means of the popularity variables between Album_types? (album, compilation, single)

I will utilize one-way MANOVA.

Variable Descriptions

Independent Variables

Album_type: the album in which the song is contained on Spotify. (album, single, compilation)
Danceability: describes how suitable a track is for dancing based on a combination of musical elements including tempo, rhythm stability, beat strength, and overall regularity.
Energy: is a measure from 0.0 to 1.0 and represents a perceptual measure of intensity and activity.
Key: the key the track is in. Integers map to pitches using standard Pitch Class notation.
Loudness: the overall loudness of a track in decibels (dB). Loudness values are averaged across the entire track and are useful for comparing relative loudness of tracks.
Speechiness: detects the presence of spoken words in a track. The more exclusively speech-like the recording (e.g. talk show, audio book, poetry), the closer to 1.0 the attribute value.
Acousticness: a confidence measure from 0.0 to 1.0 of whether the track is acoustic. 1.0 represents high confidence the track is acoustic.
Instrumentallness: predicts whether a track contains no vocals. The closer the instrumentalness value is to 1.0, the greater likelihood the track contains no vocal content.
Liveness: detects the presence of an audience in the recording. Higher liveness values represent an increased probability that the track was performed live.
Valence: tracks with high valence sound more positive (e.g. happy, cheerful, euphoric), while tracks with low valence sound more negative (e.g. sad, depressed, angry).
Tempo: the overall estimated tempo of a track in beats per minute (BPM).
Duration_ms: the duration of the track in milliseconds.

Popularity/Dependent Variables

Stream: the number of streams of the song on Spotify.
Likes: the number of likes of the song’s corresponding music video on YouTube
Views: the number of views of the YouTube video
Comments: the number of comments for the YouTube video

Exploratory Data Analysis

##   Album_type         Danceability        Energy               Key      
##  Length:20718       Min.   :0.0000   Min.   :0.0000203   Min.   : 0.0  
##  Class :character   1st Qu.:0.5180   1st Qu.:0.5070000   1st Qu.: 2.0  
##  Mode  :character   Median :0.6370   Median :0.6660000   Median : 5.0  
##                     Mean   :0.6198   Mean   :0.6352503   Mean   : 5.3  
##                     3rd Qu.:0.7402   3rd Qu.:0.7980000   3rd Qu.: 8.0  
##                     Max.   :0.9750   Max.   :1.0000000   Max.   :11.0  
##                     NA's   :2        NA's   :2           NA's   :2     
##     Loudness        Speechiness       Acousticness       Instrumentalness   
##  Min.   :-46.251   Min.   :0.00000   Min.   :0.0000011   Min.   :0.0000000  
##  1st Qu.: -8.858   1st Qu.:0.03570   1st Qu.:0.0452000   1st Qu.:0.0000000  
##  Median : -6.536   Median :0.05050   Median :0.1930000   Median :0.0000024  
##  Mean   : -7.672   Mean   :0.09646   Mean   :0.2915353   Mean   :0.0559616  
##  3rd Qu.: -4.931   3rd Qu.:0.10300   3rd Qu.:0.4772500   3rd Qu.:0.0004630  
##  Max.   :  0.920   Max.   :0.96400   Max.   :0.9960000   Max.   :1.0000000  
##  NA's   :2         NA's   :2         NA's   :2           NA's   :2          
##     Liveness         Valence           Tempo        Duration_ms     
##  Min.   :0.0145   Min.   :0.0000   Min.   :  0.0   Min.   :  30985  
##  1st Qu.:0.0941   1st Qu.:0.3390   1st Qu.: 97.0   1st Qu.: 180010  
##  Median :0.1250   Median :0.5370   Median :120.0   Median : 213284  
##  Mean   :0.1935   Mean   :0.5299   Mean   :120.6   Mean   : 224718  
##  3rd Qu.:0.2370   3rd Qu.:0.7262   3rd Qu.:139.9   3rd Qu.: 252443  
##  Max.   :1.0000   Max.   :0.9930   Max.   :243.4   Max.   :4676058  
##  NA's   :2        NA's   :2        NA's   :2       NA's   :2        
##      Views               Likes             Comments            Stream         
##  Min.   :0.000e+00   Min.   :       0   Min.   :       0   Min.   :6.574e+03  
##  1st Qu.:1.826e+06   1st Qu.:   21581   1st Qu.:     509   1st Qu.:1.767e+07  
##  Median :1.450e+07   Median :  124481   Median :    3277   Median :4.968e+07  
##  Mean   :9.394e+07   Mean   :  663341   Mean   :   27519   Mean   :1.359e+08  
##  3rd Qu.:7.040e+07   3rd Qu.:  522148   3rd Qu.:   14360   3rd Qu.:1.384e+08  
##  Max.   :8.080e+09   Max.   :50788652   Max.   :16083138   Max.   :3.387e+09  
##  NA's   :470         NA's   :541        NA's   :569        NA's   :576

`NA` values

## # A tibble: 1,169 × 16
##    Album_type Danceability Energy   Key Loudness Speechiness Acousticness
##    <chr>             <dbl>  <dbl> <dbl>    <dbl>       <dbl>        <dbl>
##  1 album             0.566  0.834     6    -8.41      0.0298     0.00277 
##  2 album             0.547  0.394     4    -9.79      0.0262     0.0522  
##  3 album             0.539  0.828     4    -9.11      0.035      0.00067 
##  4 album             0.531  0.583     7    -9.47      0.0256     0.0184  
##  5 album             0.438  0.687     7    -9.15      0.0619     0.000668
##  6 album             0.512  0.86      0    -6.14      0.0703     0.00631 
##  7 album             0.624  0.876     7    -6.00      0.0342     0.000926
##  8 album             0.404  0.266    10   -16.7       0.0334     0.716   
##  9 single            0.878  0.786    11    -5.27      0.152      0.416   
## 10 album             0.736  0.727     5    -4.64      0.0571     0.0584  
## # ℹ 1,159 more rows
## # ℹ 9 more variables: Instrumentalness <dbl>, Liveness <dbl>, Valence <dbl>,
## #   Tempo <dbl>, Duration_ms <dbl>, Views <dbl>, Likes <dbl>, Comments <dbl>,
## #   Stream <dbl>

Here are the percent NA’s for every variable

##       Album_type     Danceability           Energy              Key 
##      0.000000000      0.009653441      0.009653441      0.009653441 
##         Loudness      Speechiness     Acousticness Instrumentalness 
##      0.009653441      0.009653441      0.009653441      0.009653441 
##         Liveness          Valence            Tempo      Duration_ms 
##      0.009653441      0.009653441      0.009653441      0.009653441 
##            Views            Likes         Comments           Stream 
##      2.268558741      2.611255913      2.746404093      2.780191138

The NA values are predominantly found within the popularity variables. Since the percentages are small (all less than 3%) and are seemingly random I will remove them for easier analysis.

Popularity/Dependent Variables

As shown, there seems to be large outliers that skew the distributions and makes it difficult to see visually.

The boxplots again show that the outliers seem to be heavily influencing the distributions thus I will consider omitting them.

Omitting Outliers

The distributions are easier to visualize now; however, they are still all right skewed which will have some implications with the assumptions of the analyses and models we plan to perform.

Log-Transformation

Now I will consider a log-transformation of the popularity variables:

## Warning in bplt(at[i], wid = width[i], stats = z$stats[, i], out =
## z$out[z$group == : Outlier (-Inf) in boxplot 1 is not drawn

## Warning in bplt(at[i], wid = width[i], stats = z$stats[, i], out =
## z$out[z$group == : Outlier (-Inf) in boxplot 1 is not drawn

The distributions now look much more normal so I will consider using a log-transformation within some of our analyses.

Note: normality of predictors is not a requirement for MANOVA and regression.

Independent Variables

From the histograms we see that Acousticeness is right skewed, Liveness is right skewed, Loudness is left skewed, Speechiness is right skewed, and Duration_ms and Instrumentalness seem to be heavily influenced by extreme outliers in regards to their skewness.

Key is represeneted as a numerical variable within the dataset. It is a descrete numerical variable, but it is also a categorical variable. In music theory, the difference between the keys is not numerical but categorical thus it should be treated as a factor rather than a numeric variable within a regression model.

In regards to Duration_ms and Instrumentalness, we will not remove these extreme outliers. For duration_ms, it is plausible that there might be some very long songs. For Instrumentalness, there are songs that are completely instrumental. These can be valuable in terms of understanding the relationships between variables. Since linear regression models can be highly sensitive to outliers, we will consider using a log-transformation to minimize the impact of these extreme values.

Correlation Plot

## corrplot 0.92 loaded

From the correlation plot, we can see some trends at a glance.

Acousticness seems to be negatively correlated with several variables, especially Energy and Loudness.

Intrumentalness also seems to be negatively correlation with several variables except Acousticness.

Energy seems to have a positive correlation between Loudness, and Valence. The relationship with Loudness is predictable, however it’s interesting to see that a song with higher energy is correlated with higher positivity.

Multiple Multivariate Regression

Question: Can we predict the popularity of a song with the help of Spotify audio features?

The popularity statistics (dependent variables) would be the views, comments, and likes from its corresponding YouTube music video and streams from spotify.

##                          Views         Likes      Comments        Stream
## const             1.420745e+08  1.115131e+06  3.740722e+04  3.285848e+08
## Danceability      1.014951e+08  8.449074e+05  2.989411e+04  6.147578e+07
## Energy           -5.195799e+07 -4.466926e+05 -5.177572e+03 -1.315131e+08
## Key               4.075534e+05  2.025938e+03  2.842455e+02 -6.553343e+05
## Loudness          7.200323e+06  5.312133e+04  1.907660e+03  7.596754e+06
## Speechiness      -7.264448e+07 -1.349480e+05  1.652109e+03 -7.639498e+07
## Acousticness     -1.725573e+07 -1.748695e+05 -1.940081e+04 -8.530428e+07
## Instrumentalness -1.424082e+07 -3.946083e+04  3.961921e+03 -3.714135e+07
## Liveness         -2.341561e+07 -2.042061e+05 -1.740208e+04 -4.380464e+07
## Valence          -1.743117e+07 -3.723382e+05 -8.379393e+03 -5.766546e+07
## Tempo            -7.688880e+04 -1.554010e+02  1.153721e+01 -4.666605e+04
## Duration_ms       5.266540e+01  1.291697e-01 -7.277566e-04 -2.739933e+01

Calculating R-sq:

##      Views      Likes   Comments     Stream 
## 0.01938851 0.02339462 0.00494112 0.03038258

As shown, the R-squared values are very low.

Diagnostic Checks

Since our multivariate multiple regression model has been fit to the data, we must perform diagnostic checks for the single-response model.

We will examine the residual vectors $[\hat{\epsilon}{j1}, \hat{\epsilon}{j2}, \hat{\epsilon}{j3}, \hat{\epsilon}{j4}]$ for normality or outliers.

## # A tibble: 6 × 4
##     Obs Dependent Prediction   Residual
##   <int> <chr>          <dbl>      <dbl>
## 1     1 Views     103047048. 590508173.
## 2     1 Likes        717869.   5503027.
## 3     1 Comments      31614.    138293.
## 4     1 Stream    133226841. 907008013.
## 5     2 Views     119320444. -47308799.
## 6     2 Likes        745026.    334102.

Since our data has now been reshaped, we can plot the residual plots for each popularity metric

There are several issues with these residual plots. They appear to be biased and heteroscedastic. The model also seems to underestimate as there appears to be far more positive residuals than negative.

Log-transformation

As we saw in our exploratory data analysis, it may be necessary to consider transformations. We will log-transform Duration_ms and Instrumentalness and then transform the popularity variables.

## # A tibble: 6 × 4
##     Obs Dependent Prediction Residual
##   <int> <chr>          <dbl>    <dbl>
## 1     1 Views          16.4      3.94
## 2     1 Likes          11.7      3.97
## 3     1 Comments        8.00     4.04
## 4     1 Stream         17.5      3.25
## 5     2 Views          16.5      1.63
## 6     2 Likes          11.6      2.29

The Comments residual plot appears to show some bias with a linear trend near the bottom of the plot. The line could be the result of horizontal values for one of the variables (further analysis would be needed). However, despite the plots not being perfectly homoscedastic, these residuals plots are certainly an improvement and thus a log-transformation will continue to be used for analysis.

Checking for normality

The Q-Q plots appear to show approximate normality.

R-sq and Bootstrap Study

##      Views      Likes   Comments     Stream 
## 0.13685297 0.13620214 0.13727603 0.06582931

The R-squared values are still low; however, they are greater than the R-squared values of the non-transformed model. These values are rather impressive given the complexity behind predicting performance of a song.

Bootstrap

We will conduct a bootstrap study with 10,000 samples to create a 95% confidence interval for the R-squared values.

##            [,1]      [,2]      [,3]       [,4]
## 2.5%  0.1269251 0.1259203 0.1270456 0.05803048
## 97.5% 0.1482288 0.1478588 0.1487247 0.07535721

Bootstrap Plots

Here are the plots of the Boostrapped distributions for the R-square values. The red line indicates the actual R-square value we calculated.

Interpretation

We are 95% confident that the lower endpoint and the higher endpoint for the intervals represents the percentage of variance (when multiplied by 100) of Views, Likes, Comments, and Stream that can be explained by our model.

All of our R-sq values from the regression earlier were contained in their respective confidence intervals.

Regression Analysis

##                         Views        Likes     Comments       Stream
## const             0.116331795  0.392121615 -7.913943445 15.170056746
## Danceability      1.517486685  1.861599903  1.419443827  0.332856109
## Energy           -0.704466516 -0.839971901 -0.347103965 -1.048839393
## Key               0.010857331  0.009617646  0.009131695 -0.003657463
## Loudness          0.150321091  0.159307892  0.143565644  0.066145307
## Speechiness      -2.112394395 -1.242918042 -1.219679726 -2.045533057
## Acousticness     -0.166252350 -0.226830900 -0.574589526 -0.551188594
## Instrumentalness -1.691182704 -1.044767159 -1.152998692 -0.648563783
## Liveness         -0.128419813 -0.257898791 -0.305267855 -0.381494472
## Valence          -0.227991474 -0.794879092 -0.863124819 -0.308903935
## Tempo             0.002264389  0.002362187  0.001794977  0.001194125
## Duration_ms       1.365756885  0.976585560  1.358752413  0.320609869

Danceability seems to be positively associated with all four popularity variables (Views, Likes, Comments, Stream). It shows the largest effect on Likes.
Energy has a negative effect on all four popularity variables. It seems to have the largest negative effect on Comments.
Key shows a small positive effect on Views, Likes, and Comments, but a tiny negative effect on Stream.
Loudness is positively associated with all popularity variables, with its effect being smallest on Stream.
Speechiness negatively affects all popularity variables. It shows the largest negative effect on Stream.
Acousticness is negatively associated with all popularity variables, with its largest negative effect on Comments.
Instrumentalness has a negative effect on all popularity variables, with its largest negative effect on Comments.
Liveness has a negative effect on all popularity variables, with its largest negative effect on Comments.
Valence is negatively associated with all popularity variables, with its largest negative effect on Comments.
Tempo has a tiny positive effect on all popularity variables, with its largest positive effect on Likes.
Duration_ms shows a positive effect on Views, Likes, and Comments, but a smaller effect on Stream.

These are broad interpretations of each variable’s effect estimated effect when all other variables are held constant. The actual relationships may be more complex, especially since interactions between variables may be present.

Variable Omission

From the analysis above, we will conduct a likelihood ratio test for regression parameters to see if a subset of the predictors has a statistically significant linear relationship to the outcome.

The variables Key and Tempo have very small coefficients for all popularity variables suggesting that their contributions may be negligible.

$H_0: \beta_{Key} = \beta_{Tempo} = 0$

## [1] "Reject H0"

## [1] 0

We rejected $H_0$ which means that we cannot drop all of Key and Tempo. The result indicates that at least one of the coefficients for these variables is significantly different from zero.

We will now instead conduct the likelihood ratio test both Key and Tempo separately.

Testing $H_0: \beta_{Key} = 0$

## [1] "Reject H0"

## [1] 0.01261183

Since we rejected $H_0$ at the $\alpha = 0.05$ level, we cannot drop Key.

Now testing $H_0: \beta_{Tempo} = 0$

## [1] "Reject H0"

## [1] 6.084174e-05

The test results indicated that Key and Tempodo have a significant linear relationship with the outcomes.

Since we are using the log-transformed model, the magnitude of a coefficient doesn’t directly correspond to the magnitude of its effect on the outcome. Instead, each coefficient represents the average percentage change in the outcome for each one-unit increase in the corresponding predictor, all else being equal.

Regression Conclusion

Model Performace: The R-squared values for the non-transformed model are very low. However, the R-squared values for our log-transformed model were very reasonable given the difficulty of the research question at hand. This result suggests our model does a reasonable job at explaining variance and can decently predict a song’s popularity.

Feature Importance: Danceability, Loudness, and Duration_ms seemed to have a positive association with song popularity across all metrics (Views, Likes, Comments, Stream). Energy, Speechiness, Acousticness, Instrumentalness, Liveness, and Valence were negatively associated with song popularity across all metrics.

Omission: Key and Tempo, despite having smaller coefficients, could not be omitted from the model as they showed a statistically significant linear relationship with the outcome variables.

Improvements: Given the model performance, future research could look into the interaction effects between variables or other non-linear relationships. Additional variables not considered in this study may also contribute to song popularity.

One-Way MANOVA

Question: Are there differences in the means of the popularity variables between Album_types? (album, compilation, single)

We will compare group means with Album_type as factors.

Visualizations

Testing

$$H_0: \boldsymbol{\tau}{album} + \boldsymbol{\tau}{compilation} + \boldsymbol{\tau}_{single} = 0$$

## New names:
## Rows: 20718 Columns: 28
## ── Column specification
## ──────────────────────────────────────────────────────── Delimiter: "," chr
## (10): Artist, Url_spotify, Track, Album, Album_type, Uri, Url_youtube, T... dbl
## (16): ...1, Danceability, Energy, Key, Loudness, Speechiness, Acousticne... lgl
## (2): Licensed, official_video
## ℹ Use `spec()` to retrieve the full column specification for this data. ℹ
## Specify the column types or set `show_col_types = FALSE` to quiet this message.
## • `` -> `...1`

From the jmv package, we tried to utilize mancova, however the sample size was too large:

library(jmv)

(package <- mancova(data = df,
                    deps = vars(Views, Likes, Comments, Stream),
                    factors = Album_type))

## Error in mvnormtest::mshapiro.test(dataDeps): sample size must be between 3 and 5000

Thus we will compute MANOVA manually:

n <- nrow(df1)
g <- length(levels(as.factor(df1$Album_type)))
p <- ncol(df1[, c(13:16)])

nms <- function(data) {
  list(
    n = nrow(data),
    m = matrix(colMeans(data), ncol = 1),
    S = cov(data)
  )
}

sum_by_type <- 
  df1[, 13:16] %>%
  split(as.factor(df1$Album_type)) %>%
  map(nms)

W <- #within-group sum of squares and cross product-matrix
  sum_by_type %>%
  map(~ (.x$n - 1) * .x$S) %>%
  reduce(`+`)

xbar <- 
  sum_by_type %>%
  map(~ .x$n * .x$m) %>%
  reduce(`+`) / n

B <- #between-group sum of squares and cross-product matrix
  sum_by_type %>%
  map(~ .x$n * (.x$m - xbar) %*% t(.x$m - xbar)) %>%
  reduce(`+`)

Wilks_Lambda <- det(W) / det(W + B)

test_statistic <- ((n - p - 2) / p) * ((1 - sqrt(Wilks_Lambda)) / sqrt(Wilks_Lambda))
critical_value <- qf(0.05, 2 * p, 2 * (n - p - 2), lower.tail = FALSE)

ifelse(test_statistic > critical_value,
       "Reject H0",
       "Do not reject H0")

## [1] "Reject H0"

Wilks_Lambda

## [1] 0.9716063

We reject $H_0$ suggesting that the means between the groups of Album_type are different. Despite Wilk’s Lambda being a large value, we still rejected. This can be explained by the large sample size.

Assumptions Check

The dataset is assumed to be a random sample from from different populations that are independent.
Since our sample size for each group is large, the assumption of each population being multivariate normal can be relaxed by appealing to the central limit theorem.

Testing for equality of Covariance Matrices via the Box’s M-test:

An important assumption of MANOVA is the equality of covariance matrices across our groups. Essentially, this means we assume the variability within our groups and the relationships among our dependent variables are the same for all Album_types. We used Box’s M test to check this assumption.

$$H_0: \boldsymbol{\Sigma}{album} + \boldsymbol{\Sigma}{compilation} + \boldsymbol{\Sigma}_{single} = \boldsymbol{\Sigma}$$

## [1] 6788.335

## [1] 20

## [1] TRUE

## [1] 0

Unfortunately, we found that the assumption of equal covariance matrices was violated. This suggests that the relationships among our popularity variables may not be the same across different Album_types, which is a concern for the interpretation of our MANOVA results. Despite finding significant results in our initial MANOVA, the violation of the equal covariance matrices assumption suggests we should interpret these results with caution.

MANOVA Conclusion

Significant differences found in the popularity means across different Album_types.
Rejection of the null hypothesis according to the Wilk’s Lambda test.
Violation of equal covariance matrices assumption detected by the Box’s M-test.
Results should be interpreted with caution due to the violation of the assumption.
Additional studies are required to explore and correct for the unequal covariance matrices, possibly considering interactions, different statistical methods or data transformations.

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Multivar_report.md

Multivar_report.md

Multivariate Analysis Report

Dataset

Project Goal

Variable Descriptions

Exploratory Data Analysis

`NA` values

Popularity/Dependent Variables

Omitting Outliers

Log-Transformation

Independent Variables

Correlation Plot

Multiple Multivariate Regression

Diagnostic Checks

Log-transformation

Checking for normality

R-sq and Bootstrap Study

Bootstrap

Bootstrap Plots

Regression Analysis

Variable Omission

Regression Conclusion

One-Way MANOVA

Visualizations

Testing

Assumptions Check

Testing for equality of Covariance Matrices via the Box’s M-test:

MANOVA Conclusion

Files

Multivar_report.md

Latest commit

History

Multivar_report.md

File metadata and controls

Multivariate Analysis Report

Dataset

Project Goal

Variable Descriptions

Exploratory Data Analysis

NA values

Popularity/Dependent Variables

Omitting Outliers

Log-Transformation

Independent Variables

Correlation Plot

Multiple Multivariate Regression

Diagnostic Checks

Log-transformation

Checking for normality

R-sq and Bootstrap Study

Bootstrap

Bootstrap Plots

Regression Analysis

Variable Omission

Regression Conclusion

One-Way MANOVA

Visualizations

Testing

Assumptions Check

Testing for equality of Covariance Matrices via the Box’s M-test:

MANOVA Conclusion

`NA` values