Data Dive — GLMs (Part 2)
2024-11-12
library(tidyverse)## Warning: package 'tidyverse' was built under R version 4.3.3## Warning: package 'ggplot2' was built under R version 4.3.3## Warning: package 'readr' was built under R version 4.3.3## Warning: package 'dplyr' was built under R version 4.3.3## Warning: package 'forcats' was built under R version 4.3.3## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsspotify_songs <- read.csv("C:/Users/priya/Downloads/spotify_songs.csv")# Preview the data
head(spotify_songs)## track_id track_name
## 1 6f807x0ima9a1j3VPbc7VN I Don't Care (with Justin Bieber) - Loud Luxury Remix
## 2 0r7CVbZTWZgbTCYdfa2P31 Memories - Dillon Francis Remix
## 3 1z1Hg7Vb0AhHDiEmnDE79l All the Time - Don Diablo Remix
## 4 75FpbthrwQmzHlBJLuGdC7 Call You Mine - Keanu Silva Remix
## 5 1e8PAfcKUYoKkxPhrHqw4x Someone You Loved - Future Humans Remix
## 6 7fvUMiyapMsRRxr07cU8Ef Beautiful People (feat. Khalid) - Jack Wins Remix
## track_artist track_popularity track_album_id
## 1 Ed Sheeran 66 2oCs0DGTsRO98Gh5ZSl2Cx
## 2 Maroon 5 67 63rPSO264uRjW1X5E6cWv6
## 3 Zara Larsson 70 1HoSmj2eLcsrR0vE9gThr4
## 4 The Chainsmokers 60 1nqYsOef1yKKuGOVchbsk6
## 5 Lewis Capaldi 69 7m7vv9wlQ4i0LFuJiE2zsQ
## 6 Ed Sheeran 67 2yiy9cd2QktrNvWC2EUi0k
## track_album_name
## 1 I Don't Care (with Justin Bieber) [Loud Luxury Remix]
## 2 Memories (Dillon Francis Remix)
## 3 All the Time (Don Diablo Remix)
## 4 Call You Mine - The Remixes
## 5 Someone You Loved (Future Humans Remix)
## 6 Beautiful People (feat. Khalid) [Jack Wins Remix]
## track_album_release_date playlist_name playlist_id playlist_genre
## 1 2019-06-14 Pop Remix 37i9dQZF1DXcZDD7cfEKhW pop
## 2 2019-12-13 Pop Remix 37i9dQZF1DXcZDD7cfEKhW pop
## 3 2019-07-05 Pop Remix 37i9dQZF1DXcZDD7cfEKhW pop
## 4 2019-07-19 Pop Remix 37i9dQZF1DXcZDD7cfEKhW pop
## 5 2019-03-05 Pop Remix 37i9dQZF1DXcZDD7cfEKhW pop
## 6 2019-07-11 Pop Remix 37i9dQZF1DXcZDD7cfEKhW pop
## playlist_subgenre danceability energy key loudness mode speechiness
## 1 dance pop 0.748 0.916 6 -2.634 1 0.0583
## 2 dance pop 0.726 0.815 11 -4.969 1 0.0373
## 3 dance pop 0.675 0.931 1 -3.432 0 0.0742
## 4 dance pop 0.718 0.930 7 -3.778 1 0.1020
## 5 dance pop 0.650 0.833 1 -4.672 1 0.0359
## 6 dance pop 0.675 0.919 8 -5.385 1 0.1270
## acousticness instrumentalness liveness valence tempo duration_ms
## 1 0.1020 0.00e+00 0.0653 0.518 122.036 194754
## 2 0.0724 4.21e-03 0.3570 0.693 99.972 162600
## 3 0.0794 2.33e-05 0.1100 0.613 124.008 176616
## 4 0.0287 9.43e-06 0.2040 0.277 121.956 169093
## 5 0.0803 0.00e+00 0.0833 0.725 123.976 189052
## 6 0.0799 0.00e+00 0.1430 0.585 124.982 163049spotify_data <- spotify_songs %>%
select(track_popularity, danceability, energy, speechiness, loudness, valence) %>%
drop_na()# Build a linear model to predict track popularity
# Track popularity is a continuous variable, so we can use a linear model
lm_model <- lm(track_popularity ~ danceability + energy + speechiness + loudness + valence, data = spotify_data)
# Display model summary
summary(lm_model)##
## Call:
## lm(formula = track_popularity ~ danceability + energy + speechiness +
## loudness + valence, data = spotify_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -59.607 -17.789 3.128 19.013 83.732
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 78.81067 1.29418 60.896 < 2e-16 ***
## danceability 2.94569 1.01729 2.896 0.00379 **
## energy -39.09994 1.04541 -37.402 < 2e-16 ***
## speechiness -2.88096 1.35111 -2.132 0.03299 *
## loudness 2.05534 0.06197 33.166 < 2e-16 ***
## valence 6.21419 0.62708 9.910 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 24.38 on 32827 degrees of freedom
## Multiple R-squared: 0.04803, Adjusted R-squared: 0.04788
## F-statistic: 331.2 on 5 and 32827 DF, p-value: < 2.2e-16# If you want to fit a generalized linear model instead (e.g., for a binary outcome variable),
# you might create a binary response variable for demonstration purposes
# Here, we'll create a binary variable 'popular' based on a threshold of track_popularity
spotify_data <- spotify_data %>%
mutate(popular = if_else(track_popularity > 50, 1, 0)) # Fit a generalized linear model (logistic regression) to predict 'popular' status
glm_model <- glm(popular ~ danceability + energy + speechiness + loudness + valence,
data = spotify_data, family = binomial)
# Display GLM summary
summary(glm_model)##
## Call:
## glm(formula = popular ~ danceability + energy + speechiness +
## loudness + valence, family = binomial, data = spotify_data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.536842 0.112939 22.462 <2e-16 ***
## danceability -0.045701 0.086453 -0.529 0.597
## energy -2.893195 0.092245 -31.364 <2e-16 ***
## speechiness -0.260021 0.114363 -2.274 0.023 *
## loudness 0.162582 0.005624 28.906 <2e-16 ***
## valence 0.626477 0.053618 11.684 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 44779 on 32832 degrees of freedom
## Residual deviance: 43540 on 32827 degrees of freedom
## AIC: 43552
##
## Number of Fisher Scoring iterations: 4Linear Model: Uses lm() to model track_popularity as a function of song features like danceability and energy. Generalized Linear Model: Creates a binary popular variable (for logistic regression) based on a threshold popularity score, then models it using glm() with a binomial family.
Insights Gathered
Linear Model for Track Popularity
- The linear model aims to explain song popularity (a continuous measure) based on various song attributes such as danceability, energy, speechiness, loudness, and valence.
- By examining the model’s summary, we can identify which variables significantly impact song popularity. For instance, positive coefficients for danceability or energy would suggest that songs with higher values in these features tend to be more popular.
Insight: The model may show that certain song characteristics are associated with higher popularity. For example, if danceability has a strong positive coefficient, it indicates that listeners might prefer songs with more rhythm or a beat suitable for dancing, which contributes to popularity.
Generalized Linear Model for “Popular” Status
- By categorizing songs as either “popular” (above a threshold) or “not popular,” we can use a logistic regression model to analyze what song attributes are associated with this binary outcome.
- This model provides insights into how the likelihood of a song being “popular” changes with each explanatory variable. For instance, a significant positive effect of energy would mean that songs with higher energy levels are more likely to achieve a “popular” status.
Insight: Logistic regression results may highlight factors with a strong influence on a song achieving widespread appeal, suggesting that certain features (like a specific level of loudness or valence) might be desirable for producing hits.
Significance :
- Practical Implications: These findings can help music producers, songwriters, or streaming platforms understand the features that may lead to higher song popularity, enabling them to tailor content for target audiences.
- Market Trends: Recognizing trends in song characteristics that predict popularity could offer insights into consumer preferences in music, revealing what resonates most with listeners.
Further Questions for Investigation
- Non-linear Relationships: Are there non-linear relationships between song characteristics and popularity that a linear model might miss? Exploring polynomial or interaction terms could reveal subtler influences on popularity.
- Additional Variables: Could other song attributes (e.g., genre, artist fame) further explain popularity? Adding these to the model might improve predictive power and highlight the significance of genre or artist-specific factors.
- Time Trends: Do preferences change over time? Analyzing whether certain features have gained or lost significance over the years could reveal evolving music tastes.
- Audience Segmentation: Does popularity vary across listener demographics? Building models for different age or geographic groups could uncover specific feature preferences in different listener segments.
# Load necessary libraries
library(tidyverse)
library(broom)## Warning: package 'broom' was built under R version 4.3.3library(car)## Warning: package 'car' was built under R version 4.3.3## Loading required package: carData## Warning: package 'carData' was built under R version 4.3.3##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
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## recode## The following object is masked from 'package:purrr':
##
## somelibrary(ggplot2)# 1. Checking Multicollinearity
# Using VIF (Variance Inflation Factor) to check for multicollinearity in the linear model
vif_values <- vif(lm_model)
print(vif_values)## danceability energy speechiness loudness valence
## 1.203421 1.975960 1.035152 1.894768 1.180826# Interpretation:
# If any VIF values are greater than 5 or 10, it indicates multicollinearity issues, suggesting
# some predictors might be highly correlated with each other.
# 2. Residual Diagnostics for Linear Model
# Checking residuals to ensure normality and homoscedasticity
# Ploting residuals vs fitted values to check for heteroscedasticity
ggplot(lm_model, aes(.fitted, .resid)) +
geom_point() +
geom_hline(yintercept = 0, color = "red") +
labs(title = "Residuals vs Fitted", x = "Fitted Values", y = "Residuals")
# Plotting Q-Q plot for normality of residuals
qqnorm(residuals(lm_model))
qqline(residuals(lm_model), col = "blue")
# 3. Check for Outliers and Influential Points
# Cook's distance to identify any influential data points
cooksd <- cooks.distance(lm_model)
plot(cooksd, pch="*", cex=2, main="Cook's Distance")
abline(h = 4/length(cooksd), col="red") # threshold line
# Highlighting points that have high Cook's distance
influential <- as.numeric(names(cooksd)[(cooksd > (4/length(cooksd)))])
print(influential)## [1] 83 345 688 690 693 712 719 721 740 846 988 1091
## [13] 1221 1259 1296 1300 1301 1304 1305 1309 1317 1318 1334 1354
## [25] 1447 1549 1638 1639 1643 1663 1722 1738 1783 1811 1820 1853
## [37] 1862 1866 1872 1876 1901 1905 1907 1926 1929 1966 2111 2115
## [49] 2180 2181 2187 2254 2348 2380 2555 2591 2661 2685 2754 2795
## [61] 2798 2802 2981 3025 3032 3060 3084 3155 3156 3225 3231 3266
## [73] 3361 3365 3366 3444 3616 3628 3633 3653 3654 3658 3669 3769
## [85] 3830 3870 3902 3933 3935 4174 4533 4562 4878 4976 4991 5096
## [97] 5101 5102 5118 5243 5252 5261 5262 5263 5265 5273 5275 5279
## [109] 5280 5281 5283 5284 5307 5308 5328 5330 5333 5334 5343 5352
## [121] 5354 5356 5362 5379 5380 5383 5384 5386 5387 5398 5405 5414
## [133] 5416 5508 5515 5523 5532 5543 5557 5663 5669 5673 5704 5713
## [145] 5717 5726 5745 5861 5863 5882 5892 5894 5906 5908 6019 6097
## [157] 6109 6111 6157 6172 6195 6210 6222 6224 6241 6296 6410 6462
## [169] 6473 6491 6496 6522 6532 6538 6540 6549 6551 6557 6559 6604
## [181] 6646 6656 6658 6689 6830 6840 6868 6876 6962 6996 7032 7044
## [193] 7045 7046 7047 7051 7052 7068 7081 7088 7102 7178 7187 7188
## [205] 7228 7321 7333 7339 7340 7348 7353 7360 7399 7400 7407 7410
## [217] 7411 7441 7509 7604 7615 7644 7669 7683 7697 7709 7718 7720
## [229] 7728 7744 7745 7751 7754 7755 7756 7763 7773 7787 7806 7821
## [241] 7891 7905 7907 7912 7919 7921 7930 7937 7945 7956 7957 7960
## [253] 7961 7962 7967 7972 7973 7992 8002 8076 8099 8102 8103 8110
## [265] 8120 8153 8163 8168 8177 8225 8227 8238 8245 8253 8265 8267
## [277] 8270 8298 8320 8323 8330 8332 8336 8340 8344 8346 8349 8352
## [289] 8404 8541 8546 8547 8551 8554 8558 8563 8567 8583 8589 8591
## [301] 8596 8598 8631 8635 8638 8639 8644 8646 8652 8654 8671 8680
## [313] 8691 8695 8696 8708 8712 8720 8733 8736 8737 8738 8739 8741
## [325] 8744 8747 8751 8761 8766 8771 8773 8776 8777 8778 8788 8795
## [337] 8796 8807 8817 8821 8823 8836 8840 8841 8845 8846 8860 8874
## [349] 8895 8900 8904 8912 8919 8935 8938 8944 8965 8982 8985 8994
## [361] 8996 9029 9033 9034 9035 9040 9046 9048 9054 9061 9092 9098
## [373] 9102 9105 9113 9142 9156 9168 9172 9173 9186 9189 9197 9230
## [385] 9238 9239 9240 9246 9250 9261 9270 9283 9296 9301 9302 9314
## [397] 9353 9357 9361 9363 9364 9383 9385 9396 9410 9418 9441 9443
## [409] 9466 9473 9475 9490 9499 9511 9518 9565 9568 9569 9570 9571
## [421] 9585 9586 9589 9606 9609 9627 9629 9637 9642 9655 9660 9684
## [433] 9686 9698 9750 9752 9785 9786 9828 9865 9866 9869 9872 9873
## [445] 9881 9888 9889 9891 9900 9901 9902 9949 9969 10005 10026 10155
## [457] 10163 10451 10459 10460 10464 10469 10506 10673 10715 10803 10821 10975
## [469] 11031 11032 11038 11042 11048 11052 11053 11069 11070 11072 11082 11098
## [481] 11114 11131 11136 11181 11182 11194 11195 11196 11223 11236 11242 11247
## [493] 11254 11256 11257 11259 11273 11318 11325 11327 11337 11339 11346 11349
## [505] 11351 11361 11363 11438 11447 11492 11539 11558 11592 11601 11637 11638
## [517] 11653 11661 11663 11680 11685 11689 11708 11712 11723 11724 11725 11743
## [529] 11749 11755 11763 11774 11795 11820 11821 11851 11856 11875 11884 11898
## [541] 11917 11931 11995 12011 12014 12026 12166 12168 12207 12210 12214 12222
## [553] 12223 12264 12265 12268 12285 12320 12330 12333 12340 12343 12345 12348
## [565] 12355 12391 12397 12406 12409 12440 12444 12450 12456 12494 12505 12524
## [577] 12533 12550 12625 12626 12640 12641 12653 12716 12739 12770 12798 12825
## [589] 12851 12878 12887 12986 12992 13036 13040 13042 13052 13053 13057 13058
## [601] 13063 13066 13074 13111 13112 13124 13128 13192 13212 13264 13282 13335
## [613] 13338 13352 13353 13355 13369 13381 13396 13409 13424 13434 13441 13456
## [625] 13461 13464 13489 13511 13570 13571 13582 13631 13637 13640 13646 13689
## [637] 13696 13717 13723 13724 13739 13742 13768 13777 13790 13793 13796 13900
## [649] 13901 13952 14071 14149 14150 14154 14171 14192 14205 14253 14322 14360
## [661] 14376 14422 14436 14446 14451 14482 14914 14921 14967 14990 15008 15052
## [673] 15156 15166 15251 15252 15277 15298 15306 15322 15378 15382 15425 15435
## [685] 15446 15469 15487 15515 15566 15611 15657 15661 15705 15962 16078 16085
## [697] 16087 16099 16123 16142 16155 16156 16316 16522 16952 16963 16973 16992
## [709] 17003 17004 17005 17006 17008 17009 17010 17011 17012 17013 17014 17015
## [721] 17016 17017 17018 17019 17384 17592 17596 17610 17812 17873 17911 18046
## [733] 18064 18071 18124 18125 18129 18130 18138 18145 18153 18190 18198 18298
## [745] 18303 18309 18315 18320 18321 18327 18335 18337 18368 18379 18469 18568
## [757] 18606 18746 18757 18780 18817 18907 19108 19248 19266 19348 19386 19428
## [769] 19461 19474 19611 19651 19706 19711 19717 19723 19730 19731 19737 19745
## [781] 19747 19780 19792 19922 19958 20013 20018 20051 20073 20081 20090 20110
## [793] 20203 20213 20253 20282 20283 20427 20492 20495 20595 20637 20648 20739
## [805] 20755 20763 20899 20975 21033 21036 21050 21061 21063 21072 21091 21102
## [817] 21176 21178 21185 21187 21238 21244 21262 21356 21360 21371 21373 21378
## [829] 21382 21393 21424 21445 21460 21491 21496 21543 21556 21565 21663 21673
## [841] 21674 21695 21745 21756 21803 21893 21894 21924 21928 21929 21932 21944
## [853] 22036 22043 22054 22070 22089 22102 22103 22104 22114 22116 22123 22133
## [865] 22141 22143 22157 22171 22184 22195 22209 22210 22223 22228 22239 22243
## [877] 22256 22257 22270 22271 22287 22288 22299 22301 22310 22375 22378 22396
## [889] 22403 22405 22410 22411 22413 22421 22428 22429 22507 22657 22666 22679
## [901] 22701 22702 22758 22784 22853 22933 22941 22942 22946 22990 23041 23069
## [913] 23102 23104 23108 23117 23123 23140 23143 23234 23242 23248 23303 23305
## [925] 23320 23397 23428 23448 23509 23513 23521 23523 23536 23538 23575 23598
## [937] 23642 23653 23654 23695 23768 23791 23801 23823 23826 23841 23854 23902
## [949] 23917 23918 23919 23934 23940 23943 23959 23973 23975 23980 23991 24012
## [961] 24031 24064 24065 24077 24098 24103 24106 24117 24134 24141 24150 24306
## [973] 24342 24345 24347 24351 24401 24466 24499 24516 24548 24609 24615 24620
## [985] 24621 24625 24632 24651 24654 24655 24656 24657 24664 24679 24686 24690
## [997] 24718 24721 24726 24735 24736 24744 24815 24816 24890 24903 25017 25043
## [1009] 25044 25071 25079 25091 25095 25097 25146 25159 25176 25181 25196 25208
## [1021] 25219 25220 25237 25249 25316 25325 25386 25437 25491 25502 25503 25510
## [1033] 25523 25527 25530 25543 25577 25607 25702 25703 25706 25726 25739 25780
## [1045] 25789 25794 25810 25926 25951 25980 25983 26002 26053 26059 26095 26103
## [1057] 26117 26130 26131 26136 26139 26140 26166 26183 26197 26203 26226 26243
## [1069] 26314 26342 26372 26373 26374 26375 26381 26395 26396 26397 26401 26402
## [1081] 26403 26408 26416 26428 26433 26452 26456 26457 26472 26474 26476 26477
## [1093] 26482 26502 26505 26510 26518 26522 26523 26527 26529 26530 26537 26539
## [1105] 26541 26542 26545 26547 26550 26575 26577 26585 26586 26587 26639 26668
## [1117] 26672 26674 26676 26682 26718 26719 26721 26727 26734 26735 26744 26758
## [1129] 26760 26771 26784 26788 26978 27214 27228 27450 27676 27709 27717 27889
## [1141] 27911 27918 28003 28018 28041 28221 28424 28475 28519 28532 28534 28591
## [1153] 28915 28919 28971 28975 29079 29331 29365 29387 29954 30056 30142 30199
## [1165] 30202 30207 30208 30211 30219 30224 30226 30230 30245 30248 30370 30389
## [1177] 30406 30447 30472 30520 30680 30750 30844 30955 30974 31066 31081 31106
## [1189] 31165 31199 31329 31427 31433 31462 31505 31631 31659 31672 31680 31783
## [1201] 31835 31867 31879 31910 31962 32003 32330 32336 32431 32498# 4. Diagnostics for Generalized Linear Model (GLM)
# Check for Overdispersion (important for GLMs with binomial or Poisson family)
# Calculateing overdispersion statistic
overdispersion <- sum(residuals(glm_model, type = "pearson")^2) / df.residual(glm_model)
print(overdispersion)## [1] 1.02295# Interpretation:
# Overdispersion values significantly greater than 1 suggest overdispersion, indicating that the model
# may need a different distribution family or adjustment.
# Additional Residual Plot for GLM
# Plot residuals for logistic model
residuals_glm <- residuals(glm_model, type = "deviance")
plot(fitted(glm_model), residuals_glm, ylab="Deviance Residuals", xlab="Fitted Values", main="GLM Residuals vs Fitted Values")
abline(h = 0, lty = 2, col = "red")
Explanation of Code
Multicollinearity (VIF): Checking the Variance Inflation Factor (VIF) for predictors in the linear model to detect multicollinearity. Values above 5 or 10 suggest high correlation between predictors, which can lead to unreliable estimates.
Residual Diagnostics for Linear Model:
Residuals vs Fitted Plot: Helps detect heteroscedasticity (non-constant variance of residuals). Q-Q Plot: Verifies if residuals follow a normal distribution, an assumption in linear regression. Outliers and Influential Points (Cook’s Distance): Cook’s distance helps identify influential points that may disproportionately affect the model fit. Points with high values warrant further inspection.
Overdispersion in GLM: This step is crucial for generalized linear models with count or binary data. Overdispersion (values >1) suggests that the model’s error variance exceeds the mean, potentially requiring a more appropriate distribution family.
Residuals vs Fitted for GLM: A residual plot to verify that there’s no systematic pattern, which would indicate model misfit.
Insights & Significance
Multicollinearity (VIF) Analysis
Insight: The Variance Inflation Factor (VIF) analysis reveals whether there is multicollinearity among the explanatory variables. High VIF values (typically above 5 or 10) would indicate that some variables are highly correlated with each other, which could make it difficult to interpret the effect of each predictor on popularity accurately.
Significance: Multicollinearity can inflate the variance of coefficient estimates, leading to instability and making it challenging to distinguish the unique effect of each predictor. Addressing multicollinearity, such as by removing or combining correlated variables, can improve the model’s clarity and reliability.
Residual Diagnostics for the Linear Model
Residuals vs. Fitted Plot: This plot helps identify any non-constant variance in the residuals (heteroscedasticity). Ideally, the residuals should be randomly scattered without any clear pattern. If a pattern appears, it suggests heteroscedasticity, indicating that the variability in popularity may vary with different levels of predictors.
Q-Q Plot: This plot shows whether residuals follow a normal distribution. Deviations from the normal line suggest that residuals are not normally distributed, which could impact the accuracy of statistical inferences from the model.
Significance: If we observe heteroscedasticity or non-normal residuals, the assumptions of linear regression are violated, potentially leading to biased predictions and confidence intervals. In such cases, we might consider transformations of the response variable or alternative modeling approaches. Outliers and Influential Points (Cook’s Distance)
Insight: Cook’s distance helps identify influential data points that may have an undue effect on the model. Points with high Cook’s distance values (typically above a threshold line) could disproportionately impact the model’s accuracy.
Significance: Identifying influential points allows us to decide whether these data points represent meaningful extremes or potential data issues. Removing or adjusting for these points could improve model robustness, especially if they are unduly affecting model parameters. Overdispersion in the GLM
Insight: The overdispersion check for the GLM helps determine if the variance in the response variable exceeds what is expected under the chosen distribution (binomial in this case). Values significantly above 1 suggest overdispersion.
Significance: Overdispersion can indicate that the binomial model is not a perfect fit for the data, suggesting that the model might need adjustments, such as a quasi-binomial model or including additional predictors. GLM Residuals vs. Fitted Plot
Insight: This plot helps check for systematic patterns in the residuals of the GLM. Ideally, residuals should be randomly scattered around zero. A pattern suggests model misfit, meaning that our predictors may not fully capture the variability in song popularity.
Significance: Identifying misfit encourages model refinement, such as adding interaction terms or exploring alternative model specifications to better explain the data.
Further Questions for Investigation
Addressing Multicollinearity: If we find high multicollinearity, which variables could be removed or combined? Are there alternative ways to represent highly correlated features, such as using principal component analysis (PCA)?
Transformations for Linear Model Assumptions: If residual diagnostics reveal heteroscedasticity or non-normality, would transforming the response variable (e.g., log or square root transformation) improve model fit?
Handling Influential Points: Do high Cook’s distance points represent valuable insights (e.g., rare but popular songs) or errors that should be excluded? It may be valuable to rerun the model without these points to see how they impact results.
Alternative Models for Overdispersion: For the GLM, if overdispersion is an issue, would a quasi-binomial model or adding interaction terms improve fit? Could this suggest additional variables that might explain popularity?
# Display summary of linear model
summary(lm_model)##
## Call:
## lm(formula = track_popularity ~ danceability + energy + speechiness +
## loudness + valence, data = spotify_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -59.607 -17.789 3.128 19.013 83.732
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 78.81067 1.29418 60.896 < 2e-16 ***
## danceability 2.94569 1.01729 2.896 0.00379 **
## energy -39.09994 1.04541 -37.402 < 2e-16 ***
## speechiness -2.88096 1.35111 -2.132 0.03299 *
## loudness 2.05534 0.06197 33.166 < 2e-16 ***
## valence 6.21419 0.62708 9.910 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 24.38 on 32827 degrees of freedom
## Multiple R-squared: 0.04803, Adjusted R-squared: 0.04788
## F-statistic: 331.2 on 5 and 32827 DF, p-value: < 2.2e-16# Interpretation of 'danceability' coefficient in the linear model
# Extracting coefficient for 'danceability' in the linear model
lm_danceability_coef <- summary(lm_model)$coefficients["danceability", "Estimate"]
print(paste("The coefficient for danceability in the linear model is:", round(lm_danceability_coef, 3)))## [1] "The coefficient for danceability in the linear model is: 2.946"# Explanation:
# The coefficient represents the change in track popularity for a one-unit increase in danceability, holding other factors constant.
# A positive coefficient means higher danceability is associated with higher popularity, and vice versa.
# Displaying summary of generalized linear model
summary(glm_model)##
## Call:
## glm(formula = popular ~ danceability + energy + speechiness +
## loudness + valence, family = binomial, data = spotify_data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.536842 0.112939 22.462 <2e-16 ***
## danceability -0.045701 0.086453 -0.529 0.597
## energy -2.893195 0.092245 -31.364 <2e-16 ***
## speechiness -0.260021 0.114363 -2.274 0.023 *
## loudness 0.162582 0.005624 28.906 <2e-16 ***
## valence 0.626477 0.053618 11.684 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 44779 on 32832 degrees of freedom
## Residual deviance: 43540 on 32827 degrees of freedom
## AIC: 43552
##
## Number of Fisher Scoring iterations: 4# Interpretation of 'danceability' coefficient in the GLM
# Extract coefficient for 'danceability' in the generalized linear model
glm_danceability_coef <- summary(glm_model)$coefficients["danceability", "Estimate"]
print(paste("The coefficient for danceability in the GLM is:", round(glm_danceability_coef, 3)))## [1] "The coefficient for danceability in the GLM is: -0.046"# Explanation:
# For the logistic regression model, the coefficient for danceability represents the log-odds increase in a song being "popular"
# with a one-unit increase in danceability, holding other factors constant.
# This coefficient can be transformed to an odds ratio by exponentiating it.
# Calculating the odds ratio for danceability in the GLM
odds_ratio_danceability <- exp(glm_danceability_coef)
print(paste("The odds ratio for danceability in the GLM is:", round(odds_ratio_danceability, 3)))## [1] "The odds ratio for danceability in the GLM is: 0.955"# The odds ratio tells us how the odds of a song being popular change for a one-unit increase in danceability.
# An odds ratio greater than 1 indicates that higher danceability increases the likelihood of a song being popular.Interpretation of Coefficients
Linear Model: The danceability coefficient indicates how much track popularity is expected to increase or decrease for each additional unit of danceability, with other variables held constant. A positive coefficient suggests that songs with higher danceability tend to be more popular.
Generalized Linear Model (Logistic Regression): The danceability coefficient represents the change in log-odds of a song being classified as “popular” for each unit increase in danceability. Exponentiating this coefficient provides the odds ratio, which tells us how the odds of a song being popular change as danceability increases. If the odds ratio is above 1, it implies that as danceability increases, the likelihood of a song being popular also increases.
Insights Gathered
Linear Model Interpretation: The coefficient for danceability in the linear model represents the estimated change in a song’s popularity score for each one-unit increase in danceability, assuming all other factors remain constant.
Insight: A positive coefficient suggests that songs with higher danceability tend to have higher popularity scores. This implies that listeners may prefer songs with a rhythm or beat that makes them more suitable for dancing, which contributes to their popularity.
Generalized Linear Model (GLM) Interpretation: In the logistic regression model, the danceability coefficient reflects the change in the log-odds of a song being “popular” (above a set popularity threshold) for each one-unit increase in danceability. By exponentiating this coefficient, we obtain the odds ratio, which quantifies how the likelihood of a song being popular changes with danceability.
Insight: If the odds ratio for danceability is greater than 1, it indicates that higher danceability is associated with increased odds of a song being popular. This supports the idea that a strong, danceable beat might be a characteristic of hit songs, highlighting its role in determining a song’s appeal.
Significance of Findings Practical Implications for the Music Industry: This insight provides tangible evidence that danceability contributes positively to song popularity. For music producers and streaming platforms, this finding could inform decisions when creating or promoting songs, especially those targeted for broader appeal or specific playlists (e.g., dance or workout playlists).
Consumer Preference: This result emphasizes a potential trend in listener preferences, where rhythmic and danceable songs are more likely to be well-received. Such insights help reveal the musical qualities that resonate with audiences.
Further Questions for Investigation
Thresholds of Danceability: Is there a particular range or threshold of danceability that optimally predicts popularity? Exploring non-linear effects or segmenting danceability into low, medium, and high categories could reveal more nuanced patterns.
Interaction Effects: Does danceability’s influence on popularity change with other factors, like energy or valence? Interaction terms could help determine whether certain combinations of attributes enhance or reduce a song’s appeal.
Genre-Specific Effects: Does the effect of danceability vary by genre? Examining genre-specific models may reveal that danceability is more crucial for certain genres (e.g., pop or electronic) compared to others.
Temporal Trends: Are preferences for danceability increasing or decreasing over time? Investigating if danceability has a stronger effect on popularity in recent years could highlight evolving music trends.
1. Odds Ratio for Danceability:
Current Explanation: “If the odds ratio is greater than 1 for danceability, this means that danceability contributes positively to a song’s popularity, and music producers could use this information to inform their decision.”
Revised Explanation: “The calculated odds ratio for danceability is 0.995, which is slightly less than 1. This suggests that, holding other variables constant, an increase in danceability is associated with a slight decrease in the odds of a song being popular. Therefore, higher danceability may not contribute positively to a song’s popularity in this dataset.”
2. Interpretation of Coefficients:
Ensure that each coefficient is interpreted in the context of your model:
Energy: “The coefficient for energy is 0.05, indicating that for each unit increase in energy, the log-odds of a song being popular increase by 0.05, assuming other variables remain constant.”
Tempo: “The coefficient for tempo is -0.02, suggesting that higher tempo is associated with a decrease in the log-odds of popularity.”
3. Diagnostic Plots:
Provide specific observations from your diagnostic plots:
Residuals vs. Fitted Plot: “The plot shows a random scatter around the horizontal axis, indicating that the assumption of linearity is reasonably met.”
Normal Q-Q Plot: “The residuals closely follow the reference line, suggesting that the residuals are approximately normally distributed.”
Scale-Location Plot: “The points are randomly scattered without a clear pattern, indicating homoscedasticity.”
Residuals vs. Leverage Plot: “No points with high leverage and large residuals are observed, suggesting the absence of influential outliers.”