Regresi linier sederhana Kuliah #2-5 analisis regresi Usman Bustaman

Apa itu? Regresi Linier Sederhana

Regresi (Buku 5: Kutner, Et All P. 5) Sir Francis Galton (latter part of the 19th century): studied the relation between heights of parents and children noted that the heights of children of both tall and short parents appeared to "revert" or "regress" to the mean of the group. developed a mathematical description of this regression tendency, today's regression models (to describe statistical relations between variables).

linier Masih ingat Y=mX+B? Slope? Konstanta? Y m X B

Linier lebih lanjut… Linier dalam paramater… Persamaan Linier orde 1: Dst… (orde  pangkat tertinggi yang terdapat pada variabel bebasnya)

sederhana Relasi antar 2 variabel: 1 variabel bebas (independent variable) 1 variabel tak bebas (dependent variable) Y=mX+B? Mana variabel bebas? Mana variabel tak bebas? Y m X B

Bagaimana membangun Model Regresi Linier Sederhana Bagaimana membangun Model Regresi Linier Sederhana? Analisis/ Comment Grafik-2 Berikut:

Analisis/Comment Grafik-2 Berikut: D

Fungsi rata-2 (Mean Function) If you know something about X, this knowledge helps you predict something about Y.

Prediksi terbaik…  Bagaimana mengestimasi parameter dengan cara terbaik…

Regresi Linier

Regresi Linier ˆ Y= 𝛽 0 + 𝛽 1 𝑋 Y = b0 + b1Xi Populasi Koefisien regresi Sampel ˆ Y = b0 + b1Xi

Regresi Linier  Model Y X b b + = e Y X Yi Xi ? (the actual value of Yi) Y X b b 0 1 + = Yi i e Xi X

Regresi terbaik = minimisasi error Semua residual harus nol Minimum Jumlah residual Minimum jumlah absolut residual Minimum versi Tshebysheff Minimum jumlah kuadrat residual  OLS

Ordinary Least Square (OLS)

Assumptions Linear regression assumes that… 1. The relationship between X and Y is linear 2. Y is distributed normally at each value of X 3. The variance of Y at every value of X is the same (homogeneity of variances) 4. The observations are independent

Asumsi lebih lanjut… Alexander Von Eye & Christof Schuster (1998) Regression Analysis for Social Sciences

Asumsi lebih lanjut… Alexander Von Eye & Christof Schuster (1998) Regression Analysis for Social Sciences

Proses estimasi parameter (Drapper & Smith)

Koefisien regresi

Simbol-2 (Weisberg p. 22)

Makna koefisien regresi x = 0 ? b0 ≈ ….. b1 ≈ ….. - Tinggi vs berat badan - Nilai math vs stat - Lama sekolah vs pendptn - Lama training vs jml produksi …….

Regression Picture C B SSE SSR Variability due to x (regression) yi   x y A2 B2 C2 SST Total squared distance of observations from naïve mean of y  Total variation SSR Distance from regression line to naïve mean of y  Variability due to x (regression)   SSE Variance around the regression line  Additional variability not explained by x—what least squares method aims to minimize

explained by predictors SST (Sum Square TOTAL) Variance to be explained by predictors (SST) Y

SSE & SSR (SSR) (SSE) X Y Variance explained by X Variance NOT

explained by predictors SST = SSR + SSE Variance to be explained by predictors (SST) X Variance explained by X (SSR) Y Variance NOT explained by X (SSE)

Coefficient of Determination Koefisien Determinasi Coefficient of Determination to judge the adequacy of the regression model Maknanya: …. ?

Koefisien Determinasi

Salah paham ttg r2 R2 tinggi  prediksi semakin baik …. R2 tinggi  model regresi cocok dgn datanya … R2 rendah (mendekati nol)  tidak ada hubungan antara variabel X dan Y …

measures the strength of the linear association between two variables. Korelasi Buktikan…! Pearson Correlation…? Correlation measures the strength of the linear association between two variables.

Korelasi & Regresi 𝑺 𝒀 = 𝑺 𝒀𝒀 𝑺 𝑿 = 𝑺 𝑿𝑿

Uji parameter rls Kuliah 4

QUIZZZ…. Tuliskan pengertian anda mengenai Teorema Limit Pusat (Central Limit Theorem -- CLT) dan apa manfaatnya! Tuliskan statistik uji yang tepat dan asumsi yang digunakan untuk uji hipotesis berikut: H0: µ = 0 vs H1: µ ≠ 0 H0: µ1 = µ2 = µ3 = µ4 vs H1: setidaknya ada 1 µi = µj , i ≠ j

Review CLT Manfaat ?  Penentuan jml sampel (n)  Uji Hipotesis

Uji parameter RLS Linear regression assumes that… 1. The relationship between X and Y is linear 2. Y is distributed normally at each value of X 3. The variance of Y at every value of X is the same (homogeneity of variances) 4. The observations are independent

Uji signifikansi parameter β1 Apa artinya?

Distribusi sampling b1

b1 ~ Normal  ~ Normal H0: µ = 0 H1: µ ≠ 0 ≈

Uji koefisien regresi

Uji koefisien regresi

Confidence Interval for b1 Selang Kepercayaan koefisien regresi Confidence Interval for b1

Uji koefisien regresi

Confidence Interval for the intercept Selang Kepercayaan koefisien regresi Confidence Interval for the intercept

prediksi

Selang kepercayaan untuk E(Y)

Uji rata-2 > 2 populasi H0: µ1 = µ2 = µ3 = µ4 H1: setidaknya ada 1 µi = µj , i ≠ j  ANOVA (Analysis of Variance) Asumsi:

ANOVA Variasi Antar Sampel (Between Sample Variation) Statistik Uji: Variasi Antar Sampel (Between Sample Variation) Variasi Intra Sampel (Within Sample Variation)