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STATISTIKA INDUSTRI 2
TIN 4004
Pertemuan 9
• Outline: – Multiple Linear Regression and Correlation – Non Linear Regression
• Referensi:
– Montgomery, D.C., Runger, G.C., Applied Statistic and Probability for Engineers, 5th Ed. John Wiley & Sons, Inc., 2011.
– Walpole, R.E., Myers, R.H., Myers, S.L., Ye, K., Probability & Statistics for Engineers & Scientists , 9th Ed. Prentice Hall, 2012.
Multiple Linear Regression
• Terdiri atas lebih dari satu independent variable
• Metode yang digunakan untuk estimasi koefisien: – Least square estimation (metode kuadarat
terkecil)
– Normal equation (Persamaan Normal)
– Matrix approach (Sistem Matriks)
Multiple Linear Regression
• Terdiri atas lebih dari satu independent variable
• Metode yang digunakan untuk estimasi koefisien: – Least square estimation (metode kuadarat
terkecil)
– Normal equation (Persamaan Normal)
– Matrix approach (Sistem Matriks)
Multiple Linear Regression Penentuan Koefisien
• Least square estimation (metode kuadarat terkecil)
Multiple Linear Regression Least square estimation
• Persamaan least square:
• Least square normal equations:
LEAST SQUARE ESTIMATOR
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Multiple Linear Regression Least square estimation
• Contoh soal:
Multiple Linear Regression Least square estimation
Multiple Linear Regression Least square estimation
• Contoh soal:
Multiple Linear Regression Penentuan Koefisien
• Matrix approach (Sistem Matriks)
Model umum:
Normal Equations :
Least square Estimate of β :
Multiple Linear Regression Matrix approach
p = k + 1 p x p p x 1 p x 1
• Contoh soal
Multiple Linear Regression Matrix approach
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• Contoh soal
Multiple Linear Regression Matrix approach
Multiple Linear Regression Estimator of Variance
• Residual:
– the difference between the observation 𝑦𝑖 dengan nilai 𝑦 𝑖
–
Multiple Linear Regression Estimator of Variance
• Residual:
– Contoh soal:
Multiple Linear Regression Estimator of Variance
• Variance Estimator
Error atau Residual Sum of Squares
Multiple Linear Regression Estimator of Variance
• Variance Estimator
Contoh soal:
𝜎 2 = 𝑠2 =? ? ? ?
Multiple Linear Regression Uji Hipotesa
• Uji Nilai Individu Koefisien Regresi
– Area Penolakan:
Partial / Marginal Test
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Multiple Linear Regression Uji Hipotesa
– Contoh Soal:
– Kesimpulan: ???
Multiple Linear Regression Uji Hipotesa
• Uji Nilai Individu Koefisien Regresi
– Area Penolakan:
Partial / Marginal Test
Multiple Linear Regression Uji Hipotesa
– Contoh Soal:
– Kesimpulan: ???
Multiple Linear Regression Uji Hipotesa
• Uji Kesesuaian Model (Fitted Model Hypothesis Testing) – The ability of the entire function to predict the
true response in the range of the variables considered
– Reject 𝐻0interpret at least one regressor variable contributes significantly to the model
Multiple Linear Regression Uji Hipotesa
• Uji Kesesuaian Model (Fitted Model Hypothesis Testing)
– Menggunakan uji F
– Area Penolakan: 𝒇 > 𝒇𝜶(𝒗𝟏=𝒌,𝒗𝟐=𝒏−𝒑)
𝒇 =𝑺𝑺𝑹/𝒌
𝑺𝑺𝑬/(𝒏 − 𝒑)=
𝑺𝑺𝑹/𝒌
𝒔𝟐
Multiple Linear Regression Uji Hipotesa
• Uji Kesesuaian Model (Fitted Model Hypothesis Testing)
– Format ANOVA
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Multiple Linear Regression Confident Interval
• CI on Individual Regression Coefficients
– Contoh:
Multiple Linear Regression Confident Interval
• CI on Mean Response
– Contoh:
Multiple Linear Regression Correlation
• Coefficient of multiple determination 𝑅2
• Adjoint 𝑅2
Multiple Linear Regression Multicollinearity
• strong dependencies among regressor variables 𝑥𝑗 – The estimates of the regression coefficients are
very imprecise and affects the stability of the regression coefficients.
– To detect: • Variance inflation factors > 1
• Significant F-test of significance of regression, but tests on the individual regression coefficients are not significant
Multiple Linear Regression Uji Hipotesa
• Uji Koefisien Subset
– Test the siginificance of a set of variables. Test contribution of new variables.
– Menggunakan uji F
Partial F-test
– Area Penolakan: 𝒇 > 𝒇𝜶(𝒗𝟏=𝒓,𝒗𝟐=𝒏−𝒑)
𝒇 =𝑺𝑺𝑹(𝜷𝒋|𝜷𝟎, 𝜷𝟏, … , 𝜷𝒋−𝟏, 𝜷𝒋+𝟏, … , 𝜷𝒌)/𝒓
𝑺𝑺𝑬/(𝒏 − 𝒑)
=(𝑺𝑺𝑹 𝜷𝟏,𝜷𝟐,…,𝜷𝒌 𝜷𝟎 −𝑺𝑺𝑹 𝜷𝒋 𝜷𝟎 )/𝒓
𝒔𝟐
Multiple Linear Regression Uji Hipotesa
• Uji Koefisien Subset
– Contoh: Kasus Wire Bond Strength
𝒇 =𝟑𝟑. 𝟐/𝟐
𝟒. 𝟏= 𝟒. 𝟎𝟓
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NOTE: partial F-test to a single variable = t-test
General Linear Model (GLM)
• GLM is the mathematical framework used in many common statistical analysis, including multiple regression and ANOVA
– ANOVA is typically prsented as distinct from multiple regression but it IS a multiple regression
Characteristics of GLM
• Linear, pairs of variables are assumed to have linear relations
• Additive, if one set of variables predict another variable, the effect are thought to be additive
• BUT! This does not preclude testing non-linear or non additive effects (by doing some transformations)
Analysis of Variance (ANOVA)
• Appropriate when the predictors (independent variables) are all categorical and the outcome (dependent variable) is continous – Most common application is to analyze data from
randomized experiments
• More specifically, randomized experiments that generate more than 2 means – If only 2 means thes use:
• Independent t-test
• Dependent (paired) t-test
NONLINEAR REGRESSION
Nonlinear Regression
Beberapa Jenis Nonlinear Regression:
• Polynomial Regression Models
– Bersifat curvilinear
• Logistic Regression
– For non normal distribution data, binary responses
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TUGAS KELOMPOK
• Cari kasus permasalahan yang diselesaikan dengan: – One way ANOVA
– Factorial ANOVA
– Simple Linear Regression
– Multiple Linear Regression
• Selesaikan dengan menggunakan software statistik
• Interpretasikan hasil output software tersebut
• Catatan: – Kasus yang digunakan tidak boleh sama antar kelompok
– Tugas dipresentasikan pada pertemuan selanjutnya
Pertemuan 10 - Persiapan
• Materi Presentasi Tugas – ANOVA dan Regresi Linier: Software dan aplikasi