Multiple Regression & Two Way Anvoa by Muhammad Qasim , Aroj Bashir
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The document discusses multiple regression and two-way ANOVA. It provides definitions and examples of multiple regression, which uses two or more independent variables to predict an outcome, and two-way ANOVA, which analyzes the effects of two categorical explanatory variables. The document also provides an example problem analyzing the propellant burning rates from different missile systems and propellant types using a two-way ANOVA.
Multiple Regression & Two Way Anvoa by Muhammad Qasim , Aroj Bashir
- 1. Muhammad Qasim Aroj Bashir Dept. Environmental Sciences University of Gujrat
- 2. What is Regression What is Multi-Regression What is ANOVA & Two Way ANOVA Difference Between Multi-Regression & Two way ANOVA. Example of Multi Regression Example of Two ay ANOVA
- 3. Regression is the relationship between one dependent variable (usually denoted by Y) and a series of other changing variables (known as independent variables). two purposes of regression procedures— prediction and explanation. Example: The price of a commodity, interest rates, particular industries or sectors influence the price movement of an asset. Types of Regression: 1- Liner Regression: Linear regression uses one independent variable to explain and/or predict the outcome of Y. General Form is, Linear Regression: Y = a + bX + u
- 4. Multiple regression uses two or more independent variables to predict the outcome. Y=a + b1X1 + b2X2 + b3X3 There are several different kinds of multiple regressions—simultaneous, stepwise, and hierarchical multiple regression. In simultaneous (aka, standard) multiple regression, all the independent variables are considered at the same time.
- 5. Stepwise multiple regression In the computer determines the order in which the independent variables become part of the equation. In hierarchical multiple regression The researcher determines the order that the independent variables are entered in the equation. The order for entering the variables should be based on theory. It is possible to use nominal level data as an independent variable. For example, gender (female and male) can be used as an independent variable.
- 6. Analysis of variance is a method for splitting the total variation of our data into meaningful components that measure different source of variation. Two meaningful components 1- Measuring the variation due to experimental error 2- Measuring the variation due to experimental error plus any variation due to the different variable. Two Types: OneWay ANOVA---Single Criterion TwoWay ANOVA--- Two Criterion
- 7. Two-way (or multi-way) ANOVA is an appropriate analysis method for a study with a quantitative outcome and two (or more) categorical explanatory variables. Suppose we now have two categorical explanatory variables Is there a significant X1 effect? Is there a significant X2 effect? Are there significant interaction effects? If X1 has k levels and X2 has m levels, then the analysis is often referred to as a “k by m ANOVA” or “k x mANOVA”
- 8. S.No Multiple Regression TwoWay ANOVA 1 One or more continuous predictor variables One or more categorical variables 2 Shows relationship between variables Compares means 3 Regression technique produces adjusted F or t statistics ANOVA technique produces adjusted F statistics, and depending on the software 4 Predict the duration of child feeding in weeks using mother's age as a predictor variable Duration of child feeding in weeks using mother's marital status (single, married, divorced, widowed),
- 9. Example: In an experiment conducted to determine which of 3 missile system is preferable, the propellant burning rate for 24 static firings was measured. Four propellant type were used. The experiment yielded duplicate observations of burnings rates at each combinations of the treatments. The data, after coding, were recorded as follow? Missile System Propellant Type b1 b2 b3 b4 a1 34 30.1 29.8 29 32.7 32.8 26.7 28.9 a2 32 30.2 28.7 27.6 33.2 29.8 28.1 27.8 a3 28.4 27.3 29.7 28.8 29.3 28.9 27.3 29.1
- 10. Solution 1. Hypothesis 2. Level of Significance 3. Critical Region 4. Test Statics 5. Computations 6. Decision
- 11. Hypothesis ’: There is no difference in the mean propellant burning rate when different missile systems are used. A- H0 ’’: There is no difference in the mean propellant burning rate of the 4th propellant types. B -H0 ’’’: There is no interaction between the different missile systems and the different propellant types Level of Significance C- H0 0.05 is the level of significance Critical Region Value compression of computed f & tabulated f f > 3.13, f > 3.52, f > 2.59
- 12. Test Statics Sum of Square colum : SSC= Ƹci2/ni - (Ƹci)2/N Sum of Square of Row : SSR= Ƹri2/mi - (Ƹci)2/N Interaction : SS(RC)= Ƹ(viti)/r - (SSC+SSR+(Ƹci)2/N Sum of Square Error: SSE= SST-SSC-SSR-SS(RC) Sum of Square of Total : SST= Ƹxi2 - (Ƹci)2)/N
- 13. Computations SSC= Ƹci2/ni - (Ƹci)2/N = 126336.5/6 – 504384.04/24 =21056.08-21016.00167 = 40.0783333 Missile System Propellant Type b1 b2 b3 b4 Ƹri Ƹri2 a1 34 30.1 29.8 29 244 59536 32.7 32.8 26.7 28.9 a2 32 30.2 28.7 27.6 237.4 56358.76 33.2 29.8 28.1 27.8 a3 28.4 27.3 29.7 28.8 228.8 52349.44 29.3 28.9 27.3 29.1 Ƹci 189.6 179.1 170.3 171.2 710.2168244.2 Ƹci2 35948.16 32076.81 29002.09 29309.44 126336.5
- 14. Computations SSR= Ƹri2/mi - (Ƹci)2/N = 168244.2/8 – 504384.04/24 =21030.525-21016.00167 = 14.52333 Missile System Propellant Type b1 b2 b3 b4 Ƹri Ƹri2 a1 34 30.1 29.8 29 244 59536 32.7 32.8 26.7 28.9 a2 32 30.2 28.7 27.6 237.4 56358. 76 33.2 29.8 28.1 27.8 a3 28.4 27.3 29.7 28.8 228.8 52349. 44 29.3 28.9 27.3 29.1 Ƹci 189.6 179.1 170.3 171.2 710.2 168244 .2 Ƹci2 35948. 16 32076. 81 29002. 09 29309. 44 126336 .5
- 15. Computations SS(RC)= Ƹ(viti)/r - (SSC+SSR+(Ƹci)2/N) = 42185.54/ –(40.0783333 + 14.52333 +504384.04/2) =21092.77-21070.6 = 22.1666667 Formulated Table Missile System Propellant Type A B C D 1 66.7 62.9 56.5 57.9 2 65.2 60 56.8 55.4 3 57.7 56.2 57 57.9 Ƹ(viti)/r : 42185.54
- 16. Computations SST= Ƹxi2 - (Ƹci)2)/N = (34.02+32.72+30.12+………28.82+29.12) - 504384.04/24 =21107.68 - 21016.00167 = 91.67833 Missile System Propellant Type b1 b2 b3 b4 Ƹri Ƹri 2 a1 34 30.1 29.8 29 244 59536 32.7 32.8 26.7 28.9 a2 32 30.2 28.7 27.6 237. 4 56358. 76 33.2 29.8 28.1 27.8 a3 28.4 27.3 29.7 28.8 228. 8 52349. 44 29.3 28.9 27.3 29.1 Ƹci 189. 6 179. 1 170. 3 171. 2 710 .2 16824 4.2 Ƹci2 35948. 16 32076. 81 29002. 09 29309. 44 12633 6.5
- 17. Computation SSE= SST-SSC-SSR-SS(RC) = 91.67833- 40.0783333- 14.52333- 22.1666667 = 14.91 ANOVA TABLE Source of Variation d.f Sum of Square (SS) Mean Sequare (MS) Computed f f Crit. SSC 3 40.0783333 13.35 17.024 3.13 SSR 2 14.52333 7.261 9.254 3.52 SS(RS) 6 22.1666667 3.694 4.7024 2.63 SSE 19 14.91 0.7847 SST 24 91.67833 3.81993
- 18. ANOVA TABLE Source of Variation d.f Sum of Square (SS) Mean Sequare (MS) Computed f f Crit. SSC 3 40.0783333 13.35 17.024 3.13 SSR 2 14.52333 7.261 9.254 3.52 SS(RS) 6 22.1666667 3.694 4.7024 2.63 SSE 19 14.91 0.7847 SST 24 91.67833 3.81993 Decision Our null hypothesis accepted and mean propellant rates are used