Modello Statistico per la Previsione del Peso Neonatale
Denis Zappini
2026-07-25
Contesto Aziendale
Azienda: Neonatal Health Solutions
Obiettivo: Creare un modello statistico in grado di
prevedere con precisione il peso dei neonati alla nascita, basandosi su
variabili cliniche raccolte da tre ospedali.
Il progetto mira a
migliorare la gestione delle gravidanze ad alto rischio, ottimizzare le
risorse ospedaliere e garantire migliori risultati per la salute
neonatale.
Il progetto si inserisce all’interno di un contesto di
crescente attenzione verso la prevenzione delle complicazioni
neonatali.
La possibilità di prevedere il peso alla nascita dei
neonati rappresenta un’opportunità fondamentale per migliorare la
pianificazione clinica e ridurre i rischi associati a nascite
problematiche, come parti prematuri o neonati con basso peso.
Di
seguito, i principali benefici che questo progetto porterà all’azienda e
al settore sanitario:
1. Miglioramento delle previsioni
cliniche: ◦ Il peso del neonato è un indicatore chiave della
sua salute. Avere un modello predittivo accurato consente al personale
medico di intervenire tempestivamente in caso di anomalie, riducendo le
complicazioni perinatali come le difficoltà respiratorie o
l’ipoglicemia.
2. Ottimizzazione delle risorse
ospedaliere: ◦ Sapere in anticipo quali neonati potrebbero
avere bisogno di cure intensive aiuta a organizzare le risorse umane e
tecnologiche degli ospedali in modo efficiente. Questo si traduce in una
riduzione dei costi operativi e una migliore pianificazione
dell’utilizzo delle unità di terapia intensiva neonatale (TIN).
3.
Prevenzione e identificazione dei fattori di rischio: ◦
Il modello potrà evidenziare i fattori che maggiormente influenzano
negativamente il peso del neonato (come il fumo materno, gravidanze
multiple o età avanzata della madre). Queste informazioni sono preziose
per la prevenzione e la gestione personalizzata delle gravidanze,
permettendo interventi proattivi in caso di rischio elevato.
4.
Valutazione delle pratiche ospedaliere: ◦ Attraverso
un’analisi comparativa tra i tre ospedali coinvolti, l’azienda potrà
identificare eventuali differenze nei risultati clinici, come una
maggiore incidenza di parti cesarei in una determinata struttura. Ciò
consente di monitorare la qualità delle pratiche e armonizzare i
protocolli tra i diversi centri ospedalieri, migliorando la coerenza
delle cure.
5. Supporto alla pianificazione
strategica: ◦ L’analisi dei dati e le previsioni possono essere
utilizzate per prendere decisioni informate non solo a livello clinico
ma anche strategico. L’azienda potrà sfruttare queste informazioni per
implementare nuove politiche di salute pubblica, garantendo un impatto
positivo sui tassi di mortalità e morbilità neonatale.
Dettagli del Progetto
1. Raccolta dei Dati e Struttura del Dataset
Per costruire il modello predittivo, abbiamo raccolto dati su 2500
neonati provenienti da tre ospedali. Le variabili raccolte
includono:
• Età della madre: Misura dell’età in
anni.
• Numero di gravidanze: Quante gravidanze ha
avuto la madre.
• Fumo materno: Un indicatore
binario (0=non fumatrice, 1=fumatrice).
• Durata della
gravidanza: Numero di settimane di gestazione.
•
Peso del neonato: Peso alla nascita in grammi.
•
Lunghezza e diametro del cranio: Lunghezza del neonato
e diametro craniale, misurabili anche durante la gravidanza tramite
ecografie.
• Tipo di parto: Naturale o cesareo.
• Ospedale di nascita: Ospedale 1, 2 o 3.
•
Sesso del neonato: Maschio (M) o femmina (F).
L’obiettivo principale è identificare quali di queste variabili sono più
predittive del peso alla nascita, con un focus particolare sull’impatto
del fumo materno e delle settimane di gestazione, che potrebbero
indicare nascite premature.
2. Analisi e Modellizzazione
Analisi Preliminare
Nella prima fase, esploreremo le variabili attraverso un’analisi
descrittiva per comprenderne la distribuzione e identificare eventuali
outlier o anomalie.
Inoltre si saggeranno le seguenti ipotesi con i
test adatti:
1. In alcuni ospedali si fanno più parti
cesarei.
2. La media del peso e della lunghezza di
questo campione di neonati sono significativamente uguali a quelle della
popolazione
3. Le misure antropometriche sono
significativamente diverse tra i due sessi.
Creazione del Modello di Regressione
Verrà sviluppato un modello di regressione lineare multipla che
includa tutte le variabili rilevanti.
In questo modo, potremo
quantificare l’impatto di ciascuna variabile indipendente sul peso del
neonato ed eventuali interazioni.
Ad esempio, ci aspettiamo che una
maggiore durata della gestazione aumenterebbe in media il peso del
neonato.
Selezione del Modello Ottimale
Attraverso tecniche di selezione del modello, come la minimizzazione
del criterio di informazione di Akaike (AIC) o di Bayes (BIC),
selezioneremo il modello più parsimonioso, eliminando le variabili non
significative. Verranno considerati anche modelli con interazioni tra le
variabili e possibili effetti non lineari.
Analisi della Qualità del Modello
Una volta ottenuto il modello finale, valuteremo la sua capacità
predittiva utilizzando metriche come R² e il Root Mean Squared Error
(RMSE). Un’attenzione particolare sarà rivolta all’analisi dei residui e
alla presenza di valori influenti, che potrebbero distorcere le
previsioni, indagando su di essi.
3. Previsioni e Risultati
Una volta validato il modello, lo useremo per fare previsioni
pratiche. Ad esempio, potremo stimare il peso di una neonata
considerando una madre alla terza gravidanza che partorirà alla 39esima
settimana.
4. Visualizzazioni
Infine, utilizzeremo grafici e rappresentazioni visive per comunicare
i risultati del modello e mostrare le relazioni più significative tra le
variabili. Ad esempio, potremmo visualizzare l’impatto del numero di
settimane di gestazione e del fumo sul peso previsto.
Conclusioni
Il progetto di previsione del peso neonatale è un’iniziativa
fondamentale per Neonatal Health Solutions.
Attraverso l’uso di dati
clinici dettagliati e strumenti di analisi statistica avanzati, possiamo
contribuire a migliorare la qualità della cura prenatale, ridurre i
rischi per i neonati e ottimizzare l’efficienza delle risorse
ospedaliere.
Questo progetto rappresenta un punto di svolta per
l’azienda, consentendo non solo un miglioramento della pratica clinica
ma anche l’implementazione di politiche sanitarie più informate e
proattive.
## 'data.frame': 2500 obs. of 10 variables:
## $ Anni.madre : int 26 21 34 28 20 32 26 25 22 23 ...
## $ N.gravidanze: int 0 2 3 1 0 0 1 0 1 0 ...
## $ Fumatrici : int 0 0 0 0 0 0 0 0 0 0 ...
## $ Gestazione : int 42 39 38 41 38 40 39 40 40 41 ...
## $ Peso : int 3380 3150 3640 3690 3700 3200 3100 3580 3670 3700 ...
## $ Lunghezza : int 490 490 500 515 480 495 480 510 500 510 ...
## $ Cranio : int 325 345 375 365 335 340 345 349 335 362 ...
## $ Tipo.parto : chr "Nat" "Nat" "Nat" "Nat" ...
## $ Ospedale : chr "osp3" "osp1" "osp2" "osp2" ...
## $ Sesso : chr "M" "F" "M" "M" ...Statistical Description of the Data
## Anni.madre N.gravidanze Gestazione Peso
## Min. : 0.00 Min. : 0.0000 Min. :25.00 Min. : 830
## 1st Qu.:25.00 1st Qu.: 0.0000 1st Qu.:38.00 1st Qu.:2990
## Median :28.00 Median : 1.0000 Median :39.00 Median :3300
## Mean :28.16 Mean : 0.9812 Mean :38.98 Mean :3284
## 3rd Qu.:32.00 3rd Qu.: 1.0000 3rd Qu.:40.00 3rd Qu.:3620
## Max. :46.00 Max. :12.0000 Max. :43.00 Max. :4930
## Lunghezza Cranio
## Min. :310.0 Min. :235
## 1st Qu.:480.0 1st Qu.:330
## Median :500.0 Median :340
## Mean :494.7 Mean :340
## 3rd Qu.:510.0 3rd Qu.:350
## Max. :565.0 Max. :390Standard Deviation of the Considered Variables
## Anni.madre N.gravidanze Gestazione Peso Lunghezza Cranio
## 5.273578 1.280587 1.868639 525.038744 26.318644 16.425330Coefficient of Variation of the Considered Variables
## Anni.madre N.gravidanze Gestazione Peso Lunghezza Cranio
## 18.724536 130.512310 4.793792 15.987388 5.320208 4.830565Skewness of the Considered Variables
## Anni.madre N.gravidanze Gestazione Peso Lunghezza Cranio
## 0.0428115 2.5142541 -2.0653133 -0.6470308 -1.5146991 -0.7850527Kurtosis of the Considered Variables
## Anni.madre N.gravidanze Gestazione Peso Lunghezza Cranio
## 0.3804165 10.9894064 8.2581496 2.0315318 6.4871739 2.9462063In order to analyze these statistical variables, it is necessary to
develop the relative frequency distributions.
Wrapping up all these
information, it shall be possible to infer some considerations about the
under examination sample.
## ni fi Ni Fi
## 0 1 0.0004 1 0.0004
## 1 1 0.0004 2 0.0008
## 13 1 0.0004 3 0.0012
## 14 2 0.0008 5 0.0020
## 15 6 0.0024 11 0.0044
## 16 13 0.0052 24 0.0096
## 17 18 0.0072 42 0.0168
## 18 24 0.0096 66 0.0264
## 19 45 0.0180 111 0.0444
## 20 66 0.0264 177 0.0708
## 21 74 0.0296 251 0.1004
## 22 100 0.0400 351 0.1404
## 23 115 0.0460 466 0.1864
## 24 131 0.0524 597 0.2388
## 25 180 0.0720 777 0.3108
## 26 184 0.0736 961 0.3844
## 27 197 0.0788 1158 0.4632
## 28 172 0.0688 1330 0.5320
## 29 174 0.0696 1504 0.6016
## 30 200 0.0800 1704 0.6816
## 31 147 0.0588 1851 0.7404
## 32 159 0.0636 2010 0.8040
## 33 110 0.0440 2120 0.8480
## 34 96 0.0384 2216 0.8864
## 35 66 0.0264 2282 0.9128
## 36 64 0.0256 2346 0.9384
## 37 41 0.0164 2387 0.9548
## 38 38 0.0152 2425 0.9700
## 39 27 0.0108 2452 0.9808
## 40 19 0.0076 2471 0.9884
## 41 13 0.0052 2484 0.9936
## 42 8 0.0032 2492 0.9968
## 43 2 0.0008 2494 0.9976
## 44 4 0.0016 2498 0.9992
## 45 1 0.0004 2499 0.9996
## 46 1 0.0004 2500 1.0000
The mothers’age variable is described by a mean value of 28.16, a
standard deviation of 5.27 and, consequently, a coefficient of variation
equal to 18.72. Skewness and kurtosis are equal to, respectively, 0.04
and 0.38. It’s easy to notice that this data-set is showing two
impossible values (giving birth at 0 and 1 years old is not possible).
these data should be eliminated from the analysis.
## ni fi Ni Fi
## 0 1096 0.4384 1096 0.4384
## 1 818 0.3272 1914 0.7656
## 2 340 0.1360 2254 0.9016
## 3 150 0.0600 2404 0.9616
## 4 48 0.0192 2452 0.9808
## 5 21 0.0084 2473 0.9892
## 6 11 0.0044 2484 0.9936
## 7 1 0.0004 2485 0.9940
## 8 8 0.0032 2493 0.9972
## 9 2 0.0008 2495 0.9980
## 10 3 0.0012 2498 0.9992
## 11 1 0.0004 2499 0.9996
## 12 1 0.0004 2500 1.0000
The pregnancies number variable is described by a mean value of
0.98, a standard deviation of 1.28 and, consequently, a coefficient of
variation equal to 130.51. Skewness and kurtosis are equal to,
respectively, 2.51 and 10.99.
## ni fi Ni Fi
## 25 1 0.0004 1 0.0004
## 26 1 0.0004 2 0.0008
## 27 2 0.0008 4 0.0016
## 28 4 0.0016 8 0.0032
## 29 3 0.0012 11 0.0044
## 30 5 0.0020 16 0.0064
## 31 8 0.0032 24 0.0096
## 32 9 0.0036 33 0.0132
## 33 18 0.0072 51 0.0204
## 34 16 0.0064 67 0.0268
## 35 33 0.0132 100 0.0400
## 36 62 0.0248 162 0.0648
## 37 192 0.0768 354 0.1416
## 38 437 0.1748 791 0.3164
## 39 581 0.2324 1372 0.5488
## 40 741 0.2964 2113 0.8452
## 41 329 0.1316 2442 0.9768
## 42 56 0.0224 2498 0.9992
## 43 2 0.0008 2500 1.0000
The gestation weeks number variable is described by a mean value
of 38.98, a standard deviation of 1.87 and, consequently, a coefficient
of variation equal to 4.79. Skewness and kurtosis are equal to,
respectively, -2.06 and 8.26.
## ni fi Ni Fi
## 3300 3400 203 0.0812 203 0.0812
## 3100 3200 212 0.0848 415 0.1660
## 3600 3700 147 0.0588 562 0.2248
## 3000 3100 188 0.0752 750 0.3000
## 3500 3600 189 0.0756 939 0.3756
## 3400 3500 172 0.0688 1111 0.4444
## 2900 3000 161 0.0644 1272 0.5088
## 2300 2400 20 0.0080 1292 0.5168
## 3700 3800 131 0.0524 1423 0.5692
## 2700 2800 94 0.0376 1517 0.6068
## 3200 3300 241 0.0964 1758 0.7032
## 2600 2700 63 0.0252 1821 0.7284
## 2800 2900 144 0.0576 1965 0.7860
## 4100 4200 43 0.0172 2008 0.8032
## 3900 4000 85 0.0340 2093 0.8372
## 3800 3900 110 0.0440 2203 0.8812
## 4300 4400 20 0.0080 2223 0.8892
## 4200 4300 26 0.0104 2249 0.8996
## 2400 2500 35 0.0140 2284 0.9136
## 1300 1400 3 0.0012 2287 0.9148
## 2500 2600 48 0.0192 2335 0.9340
## 2100 2200 8 0.0032 2343 0.9372
## 4600 4700 5 0.0020 2348 0.9392
## 2200 2300 18 0.0072 2366 0.9464
## 4400 4500 9 0.0036 2375 0.9500
## 4000 4100 49 0.0196 2424 0.9696
## 2000 2100 13 0.0052 2437 0.9748
## 1400 1500 5 0.0020 2442 0.9768
## 1800 1900 4 0.0016 2446 0.9784
## 1500 1600 4 0.0016 2450 0.9800
## 1200 1300 5 0.0020 2455 0.9820
## 1700 1800 9 0.0036 2464 0.9856
## 4500 4600 7 0.0028 2471 0.9884
## 1900 2000 9 0.0036 2480 0.9920
## 1100 1200 5 0.0020 2485 0.9940
## 800 900 2 0.0008 2487 0.9948
## 1600 1700 3 0.0012 2490 0.9960
## 4800 4900 2 0.0008 2492 0.9968
## 900 1000 4 0.0016 2496 0.9984
## 4700 4800 3 0.0012 2499 0.9996
## 4900 5000 1 0.0004 2500 1.0000
The weight variable is described by a mean value of 3284 g, a
standard deviation of 525.04 g and, consequently, a coefficient of
variation equal to 15.99. Skewness and kurtosis are equal to,
respectively, -0.65 and 2.03.
## ni fi Ni Fi
## 480 490 392 0.1568 392 0.1568
## 490 500 551 0.2204 943 0.3772
## 510 520 268 0.1072 1211 0.4844
## 470 480 262 0.1048 1473 0.5892
## 500 510 392 0.1568 1865 0.7460
## 450 460 104 0.0416 1969 0.7876
## 460 470 158 0.0632 2127 0.8508
## 520 530 147 0.0588 2274 0.9096
## 440 450 58 0.0232 2332 0.9328
## 540 550 35 0.0140 2367 0.9468
## 530 540 56 0.0224 2423 0.9692
## 380 390 6 0.0024 2429 0.9716
## 390 400 4 0.0016 2433 0.9732
## 430 440 15 0.0060 2448 0.9792
## 400 410 11 0.0044 2459 0.9836
## 410 420 8 0.0032 2467 0.9868
## 350 360 4 0.0016 2471 0.9884
## 420 430 11 0.0044 2482 0.9928
## 550 560 5 0.0020 2487 0.9948
## 370 380 3 0.0012 2490 0.9960
## 300 310 1 0.0004 2491 0.9964
## 360 370 3 0.0012 2494 0.9976
## 310 320 2 0.0008 2496 0.9984
## 330 340 1 0.0004 2497 0.9988
## 320 330 1 0.0004 2498 0.9992
## 560 570 1 0.0004 2499 0.9996
## 340 350 1 0.0004 2500 1.0000
The length variable is described by a mean value of 494.7 mm, a
standard deviation of 26.32 mm and, consequently, a coefficient of
variation equal to 5.32. Skewness and kurtosis are equal to,
respectively, -1.51 and 6.49.
## ni fi Ni Fi
## 320 330 399 0.1596 399 0.1596
## 340 350 624 0.2496 1023 0.4092
## 370 380 33 0.0132 1056 0.4224
## 360 370 141 0.0564 1197 0.4788
## 330 340 658 0.2632 1855 0.7420
## 290 300 25 0.0100 1880 0.7520
## 310 320 143 0.0572 2023 0.8092
## 350 360 382 0.1528 2405 0.9620
## 380 390 11 0.0044 2416 0.9664
## 300 310 58 0.0232 2474 0.9896
## 280 290 6 0.0024 2480 0.9920
## 270 280 12 0.0048 2492 0.9968
## 250 260 2 0.0008 2494 0.9976
## 260 270 4 0.0016 2498 0.9992
## 230 240 1 0.0004 2499 0.9996
## 240 250 1 0.0004 2500 1.0000
The skull circumference variable is described by a mean value of
340 mm, a standard deviation of 16.43 and, consequently, a coefficient
of variation equal to 4.83. Skewness and kurtosis are equal to,
respectively, -0.79 and 2.95.
The variable smoker is characterized by a frequency distribution given by:
## Fumatrici
## 0 1
## 2396 104And a Gini index equal to:
## [1] 0.1594778The considered sample is characterized by a larger group of non
smoker mothers, having a Gini index indicating an almost homogeneous
group of observations..
The variable type of birth is characterized
by a frequency distribution given by:
## Tipo.parto
## Ces Nat
## 728 1772And a Gini index equal to:
## [1] 0.8256102It is possible to notice a predominant component of natural births,
even though Gini index is indicating a considerable heterogeneity and
therefore the presence of a certain number of caesarean operations. From
an epidemiological point of view, a cesarean section rate close to 29%
is a realistic figure and in line with Western hospitalization rates.
This indicates that the sample includes a main physiological component,
but also a significant share of pregnancies at risk or with
complications that required cesarean section
The variable Hospital
is characterized by a frequency distribution given by:
## Ospedale
## osp1 osp2 osp3
## 816 849 835And a Gini index equal to:
## [1] 0.9998683This variable is almost absolutely heterogeneous, having a high Gini
index. This indicates that the data-set is not affected by geographic or
structural selection bias: the data were collected in a fair and
balanced manner across the three healthcare facilities, providing a
solid basis for any future comparative analyses between
hospitals.
From the comparative analysis of the anthropometric
characteristics of newborns divided by gender, clear biological patterns
emerge:
- Newborn Weight: The median of males (M)
is visibly higher than that of females (F). The entire body distribution
(the box) of males is also shifted upward.
- Newborn Length
and Skull Diameter: Consistent with weight, both the median
length and the median skull diameter, confirm that male newborns tend to
be biologically larger and heavier at birth than females.
-
Gestation Weeks: Unlike physical measurements,
gestation length has an identical median (about 39 weeks) and a nearly
overlapping distribution between males and females. This suggests that
the gender of the newborn does not influence the length of the gestation
period, which remains regulated by purely biological factors of the
mother and standard fetal development.
A key and systematic
detail in all boxplots is the presence of a strong negative (downward)
skewness, characterized by numerous outliers (the single lower black
dots):
- Premature Births: In the Gestation Weeks
graph, there are many cases of premature births that fall below 35
weeks, reaching even borderline cases of 25 weeks.
-
Clinical Correlation of Outliers: The presence of
newborns with exceptionally low weight (under 2000g, up to less than
1000g), short lengths (under 400 mm), and small cranial diameters (under
280 mm) is the direct clinical consequence of the premature births.
Infants born significantly prematurely did not complete their
intrauterine growth, ranking as outliers in all anthropometric metrics.
- Absence of higher macro-abnormalities: On the
contrary, there are almost no higher outliers (e.g., gestations beyond
42 weeks or weights exceeding 4800 g). This suggests that the medical
protocol of the three hospitals tends to induce labor or schedule
cesarean sections before the physiological safety window is exceeded.
Hypothesis A: some hospitals perform more cesarean surgery than
others.
In order to test this affirmation, we’ll consider the following system of hypotheses: \[\left\{ \begin{array}{l} \text{H0: each hospital performs in average the same amount of caserean surgeries.}\\ \text{H1: at least one hospital performs in average a different amount of cesarean surgeries.} \end{array} \right.\]
## Tipo.parto
## Ospedale Ces Nat
## osp1 0.2965686 0.7034314
## osp2 0.2991755 0.7008245
## osp3 0.2778443 0.7221557##
## Pearson's Chi-squared test
##
## data: tab
## X-squared = 1.0972, df = 2, p-value = 0.5778Pearson’s Chi-squared test doesn’t allow to affirm a statistically
significant association between hospital and type of birth. Moreover, it
is shown a percentage of cesarean birth in the hospitals equals to
29.6%, 29.9% and 27.8% respectively. Then, the hypothesis H0 is
acceptable and there is no substantial difference in
the number of cesarean surgeries.
Hypothesis B: the average weight and length of this sample are
significantly equal to the values of the referred population.
In order to test this affirmation, we’ll consider the following system of hypotheses: \[\left\{ \begin{array}{l} \text{H0: the average weight and length of this sample are significantly equal to the values of the referred population.}\\ \text{H1: at least one between average weight and length of this sample differs significantly from the values of the referred population.} \end{array} \right.\] In a more compact form and simplifying the analysis, it is possible to consider singularly the two variables, describing two sets of hypotheses: \[\left\{ \begin{array}{l} \text{H}_0: \mu_{weight} = 3300\\ \text{H}_1: \mu_{weight} \neq 3300 \end{array} \right.\]
\[\left\{ \begin{array}{l} \text{H}_0: \mu_{length} = 500\\ \text{H}_1: \mu_{length} \neq 500 \end{array} \right.\]
##
## One Sample t-test
##
## data: Peso
## t = -1.516, df = 2499, p-value = 0.1296
## alternative hypothesis: true mean is not equal to 3300
## 95 percent confidence interval:
## 3263.490 3304.672
## sample estimates:
## mean of x
## 3284.081##
## One Sample t-test
##
## data: Lunghezza
## t = -10.084, df = 2499, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 500
## 95 percent confidence interval:
## 493.6598 495.7242
## sample estimates:
## mean of x
## 494.692Considering the obtained confidence intervals and the p-values for
both the variables, it has shown that the null hypothesis is
rejected and it is concluded that the average profile of the newborns in
the sample differs significantly from that of the population.
The standard T test performed singularly on the variables shows that
the length of the newborns differs significantly from the
average value of the population.
Hypothesis C: Anthropometric measures are significantly different
between the genders.
In this scenario, the considered variables are weight, length and
skull diameter, each of them related to the gender of the newborn. The
set of hypotheses that shall be analyzed for each variable is: \[\left\{
\begin{array}{l}
\text{H}_0: \mu_{M} = \mu_{F}\\
\text{H}_1: \mu_{M} \neq \mu_{F}
\end{array}
\right.\] T test for two independent groups could be the suitable
statistical test in order to evaluate the current hypothesis, only
though the conditions of normality and homogeneity of the variances of
the considered variables are satisfied.
Variable Weight:
##
## Shapiro-Wilk normality test
##
## data: Peso[Sesso == "M"]
## W = 0.96647, p-value = 2.321e-16##
## Shapiro-Wilk normality test
##
## data: Peso[Sesso == "F"]
## W = 0.96285, p-value < 2.2e-16##
## F test to compare two variances
##
## data: Peso by Sesso
## F = 1.136, num df = 1255, denom df = 1243, p-value = 0.02436
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
## 1.016653 1.269284
## sample estimates:
## ratio of variances
## 1.135984Variable Length:
##
## Shapiro-Wilk normality test
##
## data: Lunghezza[Sesso == "M"]
## W = 0.92028, p-value < 2.2e-16##
## Shapiro-Wilk normality test
##
## data: Lunghezza[Sesso == "F"]
## W = 0.89953, p-value < 2.2e-16##
## F test to compare two variances
##
## data: Lunghezza by Sesso
## F = 1.312, num df = 1255, denom df = 1243, p-value = 1.673e-06
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
## 1.174205 1.465987
## sample estimates:
## ratio of variances
## 1.312029Variable Skull Diameter:
shapiro.test(Cranio[Sesso=="M"])##
## Shapiro-Wilk normality test
##
## data: Cranio[Sesso == "M"]
## W = 0.97046, p-value = 3.006e-15shapiro.test(Cranio[Sesso=="F"])##
## Shapiro-Wilk normality test
##
## data: Cranio[Sesso == "F"]
## W = 0.95543, p-value < 2.2e-16var.test(Cranio~Sesso,data=dataset)##
## F test to compare two variances
##
## data: Cranio by Sesso
## F = 1.1301, num df = 1255, denom df = 1243, p-value = 0.03073
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
## 1.011431 1.262764
## sample estimates:
## ratio of variances
## 1.130149As shown in the results of the performed tests of normality and
homogeneity of the variables, T-test could head to incorrect valuations
of the hypotheses. A solution could be the Wilcoxon-Mann-Whitney non
parametric test, that offers the following results:
##
## Wilcoxon rank sum test with continuity correction
##
## data: Peso by Sesso
## W = 538640, p-value < 2.2e-16
## alternative hypothesis: true location shift is not equal to 0##
## Wilcoxon rank sum test with continuity correction
##
## data: Lunghezza by Sesso
## W = 594454, p-value < 2.2e-16
## alternative hypothesis: true location shift is not equal to 0##
## Wilcoxon rank sum test with continuity correction
##
## data: Cranio by Sesso
## W = 641638, p-value = 9.633e-15
## alternative hypothesis: true location shift is not equal to 0For each variable, the related p-value is significantly below the
usual level of 0.05. Consequently, it is possible to reject the null
hypothesis and the anthropometric measures are significantly
different between the genders.
Regression Model Creation and Selection
The variable weight isn’t described by a normal distribution. Anyway,
proceeding with the development of the model, the distribution of the
residuals shall be checked out. In fact it is more relevant than the
normality of weight.
On purpose of developing a linear regression
model comprehensive of the whole set of variables, the matrix of
correlation with the relative graphic representation shall be shown.
The first attempt of multiple linear regression model is described
by the following features:
##
## Call:
## lm(formula = Peso ~ ., data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1124.40 -181.66 -14.42 160.91 2611.89
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6738.4762 141.3087 -47.686 < 2e-16 ***
## Anni.madre 0.8921 1.1323 0.788 0.4308
## N.gravidanze 11.2665 4.6608 2.417 0.0157 *
## Fumatrici -30.1631 27.5386 -1.095 0.2735
## Gestazione 32.5696 3.8187 8.529 < 2e-16 ***
## Lunghezza 10.2945 0.3007 34.236 < 2e-16 ***
## Cranio 10.4707 0.4260 24.578 < 2e-16 ***
## Tipo.partoNat 29.5254 12.0844 2.443 0.0146 *
## Ospedaleosp2 -11.2095 13.4379 -0.834 0.4043
## Ospedaleosp3 28.0958 13.4957 2.082 0.0375 *
## SessoM 77.5409 11.1776 6.937 5.08e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 273.9 on 2489 degrees of freedom
## Multiple R-squared: 0.7289, Adjusted R-squared: 0.7278
## F-statistic: 669.2 on 10 and 2489 DF, p-value: < 2.2e-16An adjusted R-squared of 0.7278 for the first model provides a
substantial explained variability. Moreover, the considered model and
the features detected for each variable allows to suppose the
substantial effect of the variables Gestation, Length, Skull Diameter
and Gender on the Weight of the newborn. These considerations shall be
checked out in the next section through the automatic procedure
step-wise.
## Start: AIC=28139.32
## Peso ~ Anni.madre + N.gravidanze + Fumatrici + Gestazione + Lunghezza +
## Cranio + Tipo.parto + Ospedale + Sesso
##
## Df Sum of Sq RSS AIC
## - Anni.madre 1 46578 186809099 28132
## - Fumatrici 1 90019 186852540 28133
## - Ospedale 2 685979 187448501 28133
## - N.gravidanze 1 438452 187200974 28137
## - Tipo.parto 1 447929 187210450 28138
## <none> 186762521 28139
## - Sesso 1 3611021 190373542 28179
## - Gestazione 1 5458403 192220925 28204
## - Cranio 1 45326172 232088693 28675
## - Lunghezza 1 87951062 274713583 29096
##
## Step: AIC=28132.12
## Peso ~ N.gravidanze + Fumatrici + Gestazione + Lunghezza + Cranio +
## Tipo.parto + Ospedale + Sesso
##
## Df Sum of Sq RSS AIC
## - Fumatrici 1 90897 186899996 28126
## - Ospedale 2 692738 187501837 28126
## - Tipo.parto 1 448222 187257321 28130
## <none> 186809099 28132
## - N.gravidanze 1 633756 187442855 28133
## + Anni.madre 1 46578 186762521 28139
## - Sesso 1 3618736 190427835 28172
## - Gestazione 1 5412879 192221978 28196
## - Cranio 1 45588236 232397335 28670
## - Lunghezza 1 87950050 274759149 29089
##
## Step: AIC=28125.51
## Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Tipo.parto +
## Ospedale + Sesso
##
## Df Sum of Sq RSS AIC
## - Ospedale 2 701680 187601677 28119
## - Tipo.parto 1 440684 187340680 28124
## <none> 186899996 28126
## - N.gravidanze 1 610840 187510837 28126
## + Fumatrici 1 90897 186809099 28132
## + Anni.madre 1 47456 186852540 28133
## - Sesso 1 3602797 190502794 28165
## - Gestazione 1 5346781 192246777 28188
## - Cranio 1 45632149 232532146 28664
## - Lunghezza 1 88355030 275255027 29086
##
## Step: AIC=28119.23
## Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Tipo.parto +
## Sesso
##
## Df Sum of Sq RSS AIC
## - Tipo.parto 1 463870 188065546 28118
## <none> 187601677 28119
## - N.gravidanze 1 651066 188252743 28120
## + Ospedale 2 701680 186899996 28126
## + Fumatrici 1 99840 187501837 28126
## + Anni.madre 1 54392 187547285 28126
## - Sesso 1 3649259 191250936 28160
## - Gestazione 1 5444109 193045786 28183
## - Cranio 1 45758101 233359778 28657
## - Lunghezza 1 88054432 275656108 29074
##
## Step: AIC=28117.58
## Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
##
## Df Sum of Sq RSS AIC
## <none> 188065546 28118
## - N.gravidanze 1 623141 188688687 28118
## + Tipo.parto 1 463870 187601677 28119
## + Ospedale 2 724866 187340680 28124
## + Fumatrici 1 91892 187973654 28124
## + Anni.madre 1 54816 188010731 28125
## - Sesso 1 3655292 191720838 28158
## - Gestazione 1 5464853 193530399 28181
## - Cranio 1 46108583 234174130 28658
## - Lunghezza 1 87632762 275698308 29066##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio +
## Sesso, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1149.44 -180.81 -15.58 163.64 2639.72
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6681.1445 135.7229 -49.226 < 2e-16 ***
## N.gravidanze 12.4750 4.3396 2.875 0.00408 **
## Gestazione 32.3321 3.7980 8.513 < 2e-16 ***
## Lunghezza 10.2486 0.3006 34.090 < 2e-16 ***
## Cranio 10.5402 0.4262 24.728 < 2e-16 ***
## SessoM 77.9927 11.2021 6.962 4.26e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 274.6 on 2494 degrees of freedom
## Multiple R-squared: 0.727, Adjusted R-squared: 0.7265
## F-statistic: 1328 on 5 and 2494 DF, p-value: < 2.2e-16This procedure has led to a model based on the variables Number of
Pregnancies, Gestation Weeks, Length, Skull Diameter and Gender. The
adjusted R-squared of 0.7265 for the latter model is comparable to the
value of the former, equal to 0.7278. However, the latter model is based
on a fewer number of variables and, therefore, preferable.
It is
possible to compare the models using the following selection
methods:
- AIC:
## df AIC
## model_updated 7 35179.33
## model_completed 12 35171.95- BIC:
## df BIC
## model_updated 7 35220.10
## model_completed 12 35241.84
The assessment of the model requires the analysis of the
residuals, as follows:
The graphical representations seem to show that the average value
of the residuals is approximately 0, the normality of the residuals is
satisfied except for extreme values.
In order to analyze the
residuals properly, the cmean(residuals(model_updated))orrect tests
shall be performed:
Average Value of the residuals:
## [1] 3.025824e-15##
## Shapiro-Wilk normality test
##
## data: residuals(model_updated)
## W = 0.97408, p-value < 2.2e-16##
## studentized Breusch-Pagan test
##
## data: model_updated
## BP = 90.253, df = 5, p-value < 2.2e-16##
## Durbin-Watson test
##
## data: model_updated
## DW = 1.9535, p-value = 0.1224
## alternative hypothesis: true autocorrelation is greater than 0
VIF test:
## N.gravidanze Gestazione Lunghezza Cranio Sesso
## 1.023475 1.669189 2.074689 1.624465 1.040054
The mean of the residuals is 0. The Shapiro-Wilk test rejects
normality, but this is mainly due to the extreme values, corresponding
to premature births and newborns with abnormal physical characteristics.
The independence of the residuals is satisfied, but there are problems
with homoscedasticity. The analysis of the leverage and outliers values
and the Cook distance show critical issues with observation 1551. This
may be due to a data collection error, so it will be eliminated from the
data-set.
## 13 15 34 67 89 96
## 0.005630918 0.007050422 0.006744143 0.005891590 0.012816470 0.005351586
## 101 106 131 134 151 155
## 0.007526751 0.014479871 0.007229339 0.007552819 0.010883937 0.007207682
## 161 189 190 204 205 206
## 0.020335483 0.004893297 0.005366557 0.014490604 0.005351634 0.009476652
## 220 294 305 310 312 315
## 0.007393997 0.005912765 0.005442061 0.028812123 0.013169272 0.005385800
## 378 440 442 445 486 492
## 0.015934061 0.005404736 0.007723662 0.007509382 0.005164446 0.008274018
## 497 516 582 587 592 614
## 0.005166306 0.013079851 0.011665555 0.008412325 0.006384116 0.005299262
## 638 656 657 684 697 702
## 0.006688287 0.005927777 0.005322685 0.008818987 0.005863826 0.005202259
## 729 748 750 757 765 805
## 0.005023115 0.008565543 0.006942097 0.008145491 0.006070298 0.014356657
## 828 893 895 913 928 946
## 0.007179817 0.005075205 0.005295896 0.005571144 0.022742332 0.006909044
## 947 956 985 1008 1014 1049
## 0.008409465 0.007784123 0.007039416 0.005343037 0.008470133 0.004956169
## 1067 1091 1106 1130 1166 1181
## 0.008465430 0.008933360 0.005967317 0.031728597 0.005513559 0.005677676
## 1188 1200 1219 1238 1248 1273
## 0.006477203 0.005492370 0.030694311 0.005908078 0.014622914 0.007085831
## 1291 1293 1311 1321 1325 1356
## 0.006117497 0.006073639 0.009625908 0.009293111 0.004857169 0.005303442
## 1357 1385 1395 1400 1402 1411
## 0.006965051 0.012636943 0.005126697 0.005925069 0.004811441 0.008048184
## 1420 1428 1429 1450 1505 1551
## 0.005155654 0.008192811 0.021757172 0.015104831 0.013330439 0.048769569
## 1553 1556 1573 1593 1606 1610
## 0.008504889 0.005919673 0.005047204 0.005623758 0.005001812 0.008722184
## 1617 1619 1628 1686 1693 1701
## 0.004866796 0.015067498 0.005069731 0.009349313 0.005077858 0.010842957
## 1712 1718 1727 1735 1780 1781
## 0.006992084 0.006958857 0.013300523 0.004884846 0.025538678 0.016832361
## 1809 1827 1868 1892 1962 1967
## 0.008707504 0.006065698 0.005205637 0.005332812 0.005540442 0.005337356
## 1977 2037 2040 2046 2086 2089
## 0.006927281 0.004889127 0.011494872 0.005471670 0.013193090 0.006293550
## 2098 2114 2115 2120 2140 2146
## 0.005094455 0.013316875 0.011772090 0.018659995 0.006244232 0.005802168
## 2148 2149 2157 2175 2200 2215
## 0.007926839 0.013583436 0.005907225 0.032527273 0.011670024 0.004892265
## 2216 2220 2221 2224 2225 2244
## 0.008117864 0.005414040 0.021628717 0.005838076 0.005591261 0.006929217
## 2257 2307 2317 2318 2337 2359
## 0.006170254 0.013965608 0.007673614 0.004831118 0.005230450 0.010067364
## 2408 2422 2436 2437 2452 2458
## 0.009696691 0.021532808 0.004986522 0.023943328 0.023838489 0.008506087
## 2471 2478
## 0.020903740 0.005775173
OutliersTest:
## rstudent unadjusted p-value Bonferroni p
## 1551 10.051908 2.4906e-23 6.2265e-20
## 155 5.027798 5.3138e-07 1.3285e-03
## 1306 4.827238 1.4681e-06 3.6702e-03
The Bayesian Information Criterion for the regression models
selection aims to favor models characterized by a smaller number of
parameters, simplifying the model at the cost of a small loss of
information. Using this criterion and the residual analysis, the latter
model is preferable and it will be the base line for the research and
analysis of non linear effects.
In the next section, the
interaction between pairs of variables will be analyzed and,
particularly, the next two selected models:
##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio +
## Sesso, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1165.74 -179.59 -12.74 162.89 1410.88
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6683.4142 133.0802 -50.221 < 2e-16 ***
## N.gravidanze 13.1652 4.2557 3.094 0.002 **
## Gestazione 29.5891 3.7340 7.924 3.43e-15 ***
## Lunghezza 10.8927 0.3017 36.109 < 2e-16 ***
## Cranio 9.9187 0.4225 23.476 < 2e-16 ***
## SessoM 78.1348 10.9840 7.114 1.47e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 269.3 on 2493 degrees of freedom
## Multiple R-squared: 0.7372, Adjusted R-squared: 0.7367
## F-statistic: 1399 on 5 and 2493 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio +
## Sesso + Gestazione:Cranio, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1153.07 -179.27 -11.91 164.98 1457.57
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 203.85870 1084.26164 0.188 0.85088
## N.gravidanze 13.88876 4.22350 3.288 0.00102 **
## Gestazione -154.13058 28.94575 -5.325 1.10e-07 ***
## Lunghezza 11.14280 0.30182 36.919 < 2e-16 ***
## Cranio -11.72645 3.40811 -3.441 0.00059 ***
## SessoM 71.83674 10.94135 6.566 6.28e-11 ***
## Gestazione:Cranio 0.56621 0.08848 6.400 1.85e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 267.1 on 2492 degrees of freedom
## Multiple R-squared: 0.7414, Adjusted R-squared: 0.7408
## F-statistic: 1191 on 6 and 2492 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio +
## Sesso + Lunghezza:Cranio, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1169.52 -178.70 -12.43 164.69 1444.37
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.604e+02 1.009e+03 0.159 0.873652
## N.gravidanze 1.391e+01 4.219e+00 3.298 0.000987 ***
## Gestazione 3.735e+01 3.870e+00 9.650 < 2e-16 ***
## Lunghezza -3.831e+00 2.172e+00 -1.764 0.077876 .
## Cranio -1.161e+01 3.174e+00 -3.659 0.000258 ***
## SessoM 7.153e+01 1.093e+01 6.546 7.16e-11 ***
## Lunghezza:Cranio 4.429e-02 6.472e-03 6.844 9.67e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 266.8 on 2492 degrees of freedom
## Multiple R-squared: 0.742, Adjusted R-squared: 0.7414
## F-statistic: 1195 on 6 and 2492 DF, p-value: < 2.2e-16## Analysis of Variance Table
##
## Model 1: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso +
## Gestazione:Cranio
## Model 2: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2492 177817769
## 2 2493 180740178 -1 -2922409 40.956 1.854e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Analysis of Variance Table
##
## Model 1: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso +
## Lunghezza:Cranio
## Model 2: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2492 177405812
## 2 2493 180740178 -1 -3334367 46.837 9.674e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
BIC Selection Method:
## df BIC
## model_completed 12 35130.78
## model_updated 7 35107.74
## model1 8 35074.83
## model2 8 35069.03
In the next section, squared effects will be analyzed:
model_squared1 <- update(model_updated, ~ .+I(Gestazione^2)+I(Lunghezza^2)+I(Cranio^2))
summary(model_squared1)##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio +
## Sesso + I(Gestazione^2) + I(Lunghezza^2) + I(Cranio^2), data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1180.39 -180.05 -14.07 159.99 1325.41
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -8.515e+02 1.152e+03 -0.739 0.460008
## N.gravidanze 1.426e+01 4.212e+00 3.387 0.000718 ***
## Gestazione 2.571e+02 6.929e+01 3.711 0.000211 ***
## Lunghezza -1.315e+01 5.055e+00 -2.602 0.009320 **
## Cranio -1.688e+01 9.897e+00 -1.705 0.088307 .
## SessoM 7.295e+01 1.091e+01 6.688 2.77e-11 ***
## I(Gestazione^2) -2.866e+00 9.067e-01 -3.161 0.001589 **
## I(Lunghezza^2) 2.472e-02 5.144e-03 4.805 1.64e-06 ***
## I(Cranio^2) 3.974e-02 1.457e-02 2.727 0.006427 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 266.2 on 2490 degrees of freedom
## Multiple R-squared: 0.7434, Adjusted R-squared: 0.7425
## F-statistic: 901.5 on 8 and 2490 DF, p-value: < 2.2e-16model_squared2 <- update(model_updated, ~ .+I(Gestazione^2)+I(Lunghezza^2))
summary(model_squared2)##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio +
## Sesso + I(Gestazione^2) + I(Lunghezza^2), data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1187.72 -180.63 -13.61 164.12 1320.11
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.811e+03 9.019e+02 -3.117 0.001848 **
## N.gravidanze 1.440e+01 4.217e+00 3.414 0.000651 ***
## Gestazione 2.004e+02 6.617e+01 3.028 0.002489 **
## Lunghezza -1.928e+01 4.535e+00 -4.252 2.20e-05 ***
## Cranio 1.009e+01 4.202e-01 24.023 < 2e-16 ***
## SessoM 7.340e+01 1.092e+01 6.721 2.22e-11 ***
## I(Gestazione^2) -2.135e+00 8.673e-01 -2.462 0.013890 *
## I(Lunghezza^2) 3.093e-02 4.618e-03 6.697 2.62e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 266.6 on 2491 degrees of freedom
## Multiple R-squared: 0.7426, Adjusted R-squared: 0.7419
## F-statistic: 1027 on 7 and 2491 DF, p-value: < 2.2e-16model_squared3 <- update(model_updated, ~ .+I(Lunghezza^2))
summary(model_squared3)##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio +
## Sesso + I(Lunghezza^2), data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1176.79 -178.89 -12.52 164.19 1327.44
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.607e+03 7.586e+02 -2.119 0.034207 *
## N.gravidanze 1.420e+01 4.220e+00 3.364 0.000781 ***
## Gestazione 3.773e+01 3.890e+00 9.700 < 2e-16 ***
## Lunghezza -1.172e+01 3.342e+00 -3.508 0.000459 ***
## Cranio 1.015e+01 4.201e-01 24.157 < 2e-16 ***
## SessoM 7.222e+01 1.092e+01 6.614 4.57e-11 ***
## I(Lunghezza^2) 2.330e-02 3.430e-03 6.795 1.35e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 266.8 on 2492 degrees of freedom
## Multiple R-squared: 0.742, Adjusted R-squared: 0.7413
## F-statistic: 1194 on 6 and 2492 DF, p-value: < 2.2e-16anova(model_squared1,model_updated)## Analysis of Variance Table
##
## Model 1: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso +
## I(Gestazione^2) + I(Lunghezza^2) + I(Cranio^2)
## Model 2: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2490 176494079
## 2 2493 180740178 -3 -4246099 19.968 8.682e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(model_squared2,model_updated)## Analysis of Variance Table
##
## Model 1: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso +
## I(Gestazione^2) + I(Lunghezza^2)
## Model 2: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2491 177021363
## 2 2493 180740178 -2 -3718815 26.165 5.68e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(model_squared3,model_updated)## Analysis of Variance Table
##
## Model 1: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso +
## I(Lunghezza^2)
## Model 2: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2492 177452060
## 2 2493 180740178 -1 -3288119 46.176 1.348e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
BIC Selection Method:
## df BIC
## model_squared1 10 35071.81
## model_squared2 9 35071.44
## model_squared3 8 35069.69
## model_updated 7 35107.74
Combining the information obtained from the analysis of the
various models related to the interactions between variables and the
quadratic effects, some definitive models will be proposed and
analyzed.
##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Sesso +
## I(Lunghezza^2) + Lunghezza:Cranio, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1174.0 -178.5 -11.8 163.5 1377.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.879e+02 7.599e+02 -0.642 0.520884
## N.gravidanze 1.411e+01 4.218e+00 3.346 0.000831 ***
## Gestazione 3.801e+01 3.886e+00 9.783 < 2e-16 ***
## Lunghezza -9.350e+00 3.329e+00 -2.809 0.005011 **
## SessoM 7.155e+01 1.091e+01 6.556 6.72e-11 ***
## I(Lunghezza^2) 1.377e-02 3.418e-03 4.027 5.82e-05 ***
## Lunghezza:Cranio 2.070e-02 8.537e-04 24.245 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 266.7 on 2492 degrees of freedom
## Multiple R-squared: 0.7423, Adjusted R-squared: 0.7417
## F-statistic: 1197 on 6 and 2492 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Sesso +
## I(Gestazione^2) + I(Lunghezza^2) + Lunghezza:Cranio, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1186.40 -178.71 -13.79 163.75 1318.32
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.857e+03 9.032e+02 -2.056 0.039918 *
## N.gravidanze 1.432e+01 4.213e+00 3.400 0.000684 ***
## Gestazione 2.223e+02 6.606e+01 3.365 0.000778 ***
## Lunghezza -1.794e+01 4.527e+00 -3.962 7.64e-05 ***
## SessoM 7.286e+01 1.091e+01 6.679 2.96e-11 ***
## I(Gestazione^2) -2.419e+00 8.658e-01 -2.794 0.005244 **
## I(Lunghezza^2) 2.246e-02 4.618e-03 4.863 1.23e-06 ***
## Lunghezza:Cranio 2.061e-02 8.531e-04 24.157 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 266.3 on 2491 degrees of freedom
## Multiple R-squared: 0.7431, Adjusted R-squared: 0.7424
## F-statistic: 1029 on 7 and 2491 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = Peso ~ N.gravidanze + Gestazione + Lunghezza + Sesso +
## I(Gestazione^2) + I(Lunghezza^2) + Gestazione:Cranio, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1183.07 -181.38 -12.68 164.74 1315.26
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.769e+03 9.024e+02 -1.960 0.050100 .
## N.gravidanze 1.442e+01 4.207e+00 3.427 0.000620 ***
## Gestazione 1.846e+02 6.606e+01 2.794 0.005252 **
## Lunghezza -1.534e+01 4.518e+00 -3.396 0.000695 ***
## SessoM 7.280e+01 1.090e+01 6.682 2.91e-11 ***
## I(Gestazione^2) -3.085e+00 8.642e-01 -3.569 0.000364 ***
## I(Lunghezza^2) 2.690e-02 4.607e-03 5.839 5.95e-09 ***
## Gestazione:Cranio 2.634e-01 1.084e-02 24.307 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 266 on 2491 degrees of freedom
## Multiple R-squared: 0.7437, Adjusted R-squared: 0.743
## F-statistic: 1033 on 7 and 2491 DF, p-value: < 2.2e-16## Analysis of Variance Table
##
## Model 1: Peso ~ N.gravidanze + Gestazione + Lunghezza + Sesso + I(Lunghezza^2) +
## Lunghezza:Cranio
## Model 2: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2492 177205699
## 2 2493 180740178 -1 -3534480 49.705 2.305e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Analysis of Variance Table
##
## Model 1: Peso ~ N.gravidanze + Gestazione + Lunghezza + Sesso + I(Gestazione^2) +
## I(Lunghezza^2) + Lunghezza:Cranio
## Model 2: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2491 176652058
## 2 2493 180740178 -2 -4088121 28.824 4.214e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Analysis of Variance Table
##
## Model 1: Peso ~ N.gravidanze + Gestazione + Lunghezza + Sesso + I(Gestazione^2) +
## I(Lunghezza^2) + Gestazione:Cranio
## Model 2: Peso ~ N.gravidanze + Gestazione + Lunghezza + Cranio + Sesso
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2491 176234018
## 2 2493 180740178 -2 -4506160 31.846 2.204e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
BIC Selection Method:
## df BIC
## model_completed 12 35130.78
## model1 8 35074.83
## model2 8 35069.03
## model_squared1 10 35071.81
## model_squared2 9 35071.44
## model_squared3 8 35069.69
## model_optimal1 8 35066.21
## model_optimal2 9 35066.22
## model_optimal3 9 35060.30
## model_updated 7 35107.74
AIC Selection Method:
## df AIC
## model_completed 12 35060.90
## model1 8 35028.24
## model2 8 35022.45
## model_squared1 10 35013.57
## model_squared2 9 35019.02
## model_squared3 8 35023.10
## model_optimal1 8 35019.63
## model_optimal2 9 35013.81
## model_optimal3 9 35007.88
## model_updated 7 35066.98
The value of BIC method, the adjusted R-squared, the RSE and the
significance of the regressors suggest to choose the Model_Optimal 3,
based on the variables Number of Pregnancies, Gestation Weeks, Length,
Gender, square Gestation weeks, square Length and the interaction
between Gestation and Skull Diameter. The AIC method suggests the same
choice.
Residual Analysis
Test on the Residuals:
## [1] -5.44255e-15##
## Shapiro-Wilk normality test
##
## data: residuals(model_optimal3)
## W = 0.99004, p-value = 3.692e-12##
## studentized Breusch-Pagan test
##
## data: model_optimal3
## BP = 12.744, df = 7, p-value = 0.0786##
## Durbin-Watson test
##
## data: model_optimal3
## DW = 1.954, p-value = 0.125
## alternative hypothesis: true autocorrelation is greater than 0
Leverage:
## 15 34 36 61 67 89
## 0.009217556 0.007072619 0.007750389 0.007258620 0.009219682 0.012861826
## 101 106 119 131 134 151
## 0.020841255 0.023966130 0.006630263 0.007660805 0.007746642 0.012064542
## 155 161 204 206 220 304
## 0.015410273 0.023286854 0.014728272 0.013690093 0.008586739 0.010768317
## 305 310 312 317 322 328
## 0.008647797 0.089347283 0.046966660 0.007365411 0.006651253 0.007892711
## 378 383 442 445 471 486
## 0.062634450 0.007996679 0.008248747 0.008813195 0.010765751 0.007734879
## 492 513 516 582 587 592
## 0.025188603 0.006438497 0.013445055 0.012439613 0.023095804 0.012628402
## 629 638 666 684 702 706
## 0.007348826 0.006854756 0.009295664 0.009258344 0.008559252 0.006450901
## 729 748 750 757 765 805
## 0.010128111 0.016767855 0.014581749 0.008243313 0.008407510 0.036092806
## 828 895 928 946 947 949
## 0.007206700 0.007548147 0.135475686 0.008143249 0.014071937 0.006555167
## 956 958 964 985 988 991
## 0.010223005 0.006940094 0.008201226 0.008788036 0.007086213 0.007971271
## 1014 1067 1091 1130 1188 1219
## 0.035324324 0.016272783 0.009269083 0.036436029 0.009984601 0.030758760
## 1222 1238 1248 1267 1273 1275
## 0.006417583 0.007735465 0.026666340 0.006508264 0.013926381 0.008026026
## 1283 1291 1311 1321 1323 1357
## 0.007874914 0.006524212 0.009655575 0.009697146 0.009039267 0.014408171
## 1385 1400 1411 1428 1429 1450
## 0.043408332 0.009285459 0.008215317 0.037373365 0.039826400 0.015375525
## 1505 1513 1553 1556 1573 1593
## 0.013362758 0.010220782 0.008759626 0.006844490 0.006885186 0.008207740
## 1610 1619 1628 1634 1639 1686
## 0.014120951 0.079223260 0.010671824 0.008432814 0.008202322 0.020379750
## 1701 1712 1718 1727 1780 1781
## 0.021644645 0.007448235 0.009098523 0.013477920 0.130252482 0.017138223
## 1809 1893 1920 1977 2016 2040
## 0.010177435 0.006716610 0.007889417 0.009548541 0.006930662 0.023132814
## 2086 2087 2089 2114 2115 2120
## 0.013228317 0.008407025 0.010123838 0.046284483 0.026196017 0.070893140
## 2140 2148 2149 2157 2175 2200
## 0.007264059 0.008136580 0.022488744 0.006444987 0.060839031 0.025672673
## 2216 2221 2244 2257 2307 2317
## 0.020997713 0.021652488 0.006980277 0.013076293 0.026407662 0.008165337
## 2359 2391 2408 2422 2437 2452
## 0.010739079 0.012821750 0.013769331 0.021572049 0.101080177 0.093526575
## 2458 2471 2478
## 0.019715823 0.021254752 0.012381264
Outliers:
## rstudent unadjusted p-value Bonferroni p
## 1306 4.973830 7.0115e-07 0.0017522
## 155 4.948915 7.9613e-07 0.0019895
## 1399 -4.470978 8.1352e-06 0.0203300
## 1694 4.372693 1.2777e-05 0.0319290
The mean of the residuals is almost equal to 0. The Shapiro-Wilk
test rejects normality, but this is mainly due to the extreme values,
corresponding to premature births and newborns with abnormal physical
characteristics and anticipated births. The independence of the
residuals is satisfied, and homoscedasticity too. The analysis of the
leverage and outliers values and the Cook distance do not show critical
issues. The selected model will be appropriate for performing
predictions in most common situations.
Prediction and Graphics
Example of weight prediction for a male-child under these conditions:
mother with three previous pregnancies, 39 weeks of gestation, length of
530 mm and skull diameter of 330 mm.
The estimated weight is:
## 1
## 3666.444Another case about a female-child under the following conditions: no
previous pregnancy for the mother, 37 weeks of gestation, length of 500
mm and skull diameter of 300 mm.
The estimated weight is:
## 1
## 2813.123## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
Conclusion
The selected model for the interpretation of the data and performing
previsions is Model Optimal 3, based on the variables Number of
Pregnancies, Gestation Weeks, Length, Gender, square Gestation weeks,
square Length and the interaction between Gestation and Skull Diameter.
This model and the related main variables could suggest that the Age of
the Mother, the Smoking Condition, the kind of Birth (natural or
cesarean) and the Hospital are statistically insignificant on the Weight
of the newborn.
The analysis of the forecasts and residual
diagnostic plots confirms the goodness of fit of the model. Possible
critical issues are the non-total normality of the residuals (as
discussed in the previous section), the presence of outlier and leverage
values (altering the determination of the coefficients) and the omission
of variables related to the condition of the mother (for example
maternal nutrition, gestational pathologies, maternal body mass
index…).
By analyzing these issues and integrating the model
accordingly, also considering generalized linear models or non-linear
transformations, it will be possible to better manage the deviation from
normality of the residuals and reduce the unexplained variance.