Design and Analysis of Experiments Final Report of Project Group 6 孟凡兴 刘鑫鑫 许智宇 汤兆畑 刘 阳 2009-6-8

Experiment Methodology Experiment Analysis Agenda Research Background Experiment Methodology Experiment Analysis Regression Model and Prediction Strength and Weakness

Research Background Research objectives Brief introduction of the project Preparations of the project Preparation for material Implementation for measurement

Research Objectives Quantitatively study the effect of several factors on the average longest distance that the boat will travel, including the boat length, wide, relative engine location, and mechanical the engine material and the depth of the water.

The Cause and Effect Diagram Important

Brief Introduction Input Length of boat Width of boat Process Ambient temperature Length between cross bars Location of the rubber band Material Length of the paddle Turbidity of the water Width of the paddle Depth of water Noise factors Controllable factors Travel distance of the boat

Preparation of the Project Material Wood Rubber band Glue Implementation of measurements

Experiment Methodology Reference Design of Experiment

Reference Methodology Defining Parameters Full Factorial Design Regression Analysis Prediction Design of experiments for synthesizing in situ Ni–SiO2 and Co–SiO2 nanocomposites by non-isothermal reduction treatment

Parameter Definition Response variable Control variables The travel length of the boat Control variables A: Boat Length B: Boat Width C: Horizontal Distance D: Paddle Length E: Paddle Width Design of Experiment

Parameter Definition Coded Values of Variables +1 20 10 12 6 3 16 7.5 Boat Length B: Width C: Horizontal Distance D: Paddle E: Coded Values Actual +1 20 10 12 6 3 16 7.5 5 2 -1 8 1

Full Factorial Design Five-Factor two Levels design Four Center points Five-Factor two Level design Four Center points StdOrder RunOrder CenterPt Blocks A B C D E 1 42 -1 2 26 3 24 4 68 5 62 6 16 7 25 8 39 9 13 10 67 11 31 12 30 32 14 33 15 20 17 53 18 48 19 59 21 35 22 23 37 34 46 27 54 28 40 29 50 44 41 Design of Experiment

Experiment Analysis Form initial model Perform statistical testing Refine model Analyze residuals Interpret results 在实验的基本设计完成之后，下面我们就要对实验进行深入的分析。在分析中，我们采取以下分析步骤。

Form Initial Model Full model: Replicated design Linearity assumption Main effects and interactions 首先构造初始模型,我们采用的是有重复的全因子设计，我们假设模型是线性的，在这里我们主要考虑变量的主效应和交互效应。

Perform Statistical Testing Normal effect diagram Main effect and interaction effect are significant Main effect diagram Linearity is not satisfied Residual plots Outliers and pattern are observed 构造初始模型以后，我们对模型进行统计检验。首先通过正态图可以看到，主效应和交互效应都是非常显著的；通过通过主效应图可以看到，我们线性的假设是不成立的，在参差图中也出现了一些异常情况，

Perform Statistical Testing(Cont’d) Center point is significant, linearity assumption cannot be satisfied. Quadratic effects should be added.

Refine Model_1 Second-order response surface model Quadratic effects are included Quadratic effects

Perform Statistical Testing Significance level: 0.05 Main effects: A, B, C, D, E Interaction effects AD, AE, BD, DE Quadratic effects BB, CC, DD, EE

Residual Analysis An outliner P-Value=0.266 Outliner should be remove

Refine Model_2 P-Values are improved Coefficients are different after refined Before refined After refined

Residual Analysis_2 Normal Probability Plot Normality assumption Residuals Versus Fitted Values Nonconstant variance Error and background noise is not significant Residuals Versus Sequence No correlation is detected Independence assumption

Interpret Results Curvature exists A, B, C, E: center point D: high level

Interpret Results(Cont’d)

Regression Model and Prediction

Regression Model Regression model with coded values Transform between coded & uncoded values Regression model with uncoded values Coded Variables Coefficient Constant 66.5172 A 2.556 B -6.7325 C -2.5119 D 23.659 E 11.3354 B*B 6.7242 C*C -8.5258 D*D -8.0258 E*E -6.0258 A*D 2.3876 A*E 2.2314 B*D -1.8251 C*E 1.7061 D*E 4.2001 Uncoded Variables Coefficient Constant 66.5172 X1 0.639 X2 -2.693 X3 -1.25595 X4 15.77267 X5 11.3354 X2X2 1.075872 X3X3 -2.13145 X4X4 -3.56702 X5X5 -6.0258 X1X4 0.397933 X1X5 0.55785 X2X4 -0.48669 X3X5 0.85305 X4X5 2.800067 前面我们通过定性的角度分析了各个参数对小船滑行距离的影响，下面我们将通过回归方程的方法来对模型进行定量分析。 通过P-value的检验，我们可以得到各个影响显著的变量以及相应的系数，这些变量都是编码的，也就是说变量的取值只能是+1，-1和0.通过编码变量与非编码变量之间的关系，我们可以得到非编码变量的各个系数。这里面X1.X2等是非编码变量，也就是说他们的可以在规定的范围内任意取值。所以这第二个回归模型的应用会更加广泛。

Prediction Prediction with regression model of uncoded variables ID A Distance_Pre Distance_Exp 1 18 8 9 5 2 73.46 68 14 11 66.90 69 3 16 4 54.82 58 15 10 59.34 56 12 32.80 34 6 77.15 80 7 65.46 62 53.22 50 83.69 76 使用这种未编码变量的回归模型，我们对小船滑行的距离进行了九组预测。前面是预测使用的参数设计，后面是通过模型的预测值以及实验测得的真实值。那么下面我们对这两组数据进行统计检验，来验证预测的准确性。

Prediction(Cont’d) Paired t-Test Conclusion Regression model fits well Normality test (0.05) Hypothesis test Conclusion P-value=0.278>0.05 Cannot reject null hypothesis Regression model fits well 我们采用的是paired t test，在对两组数据进行正态性检验后，我们发现两组数据的正态性都非常好，通过t检验我们看到，P-value的值是0.278，我们不能拒绝两组数据相等的原假设，因此说明我们模型预测性良好，回归模型拟合的非常好。

Strength and Weakness Strength Full factorial design 64 basic samples Two replications Four center points Little influence of noise factors Temperature, air velocity Especially human factor noises 以上就是我们project的总体设计。从整体来看，我们实验的设计主要有以下两方面优势：首先，我们采用了有两次重复的全因子设计，共收集了64个基本数据，同时还有4个中心点。大量的数据有效的减少了实验中带来的随机误差，同时中心点有利于我们发现模型的非线性，这样使实验的结果更加准确。 第二方面，我们实验收到噪音因素的影响非常少。因为我们的实验不会收到温度、气流等不可控因素的影响，更重要的是我们减少了人的影响，收集的收据更加准确。

Strength and Weakness(Cont’d) Reading error Boat making error Other uncontrollable factors Water depth Pool width Paddle position 但同时，我们的实验还存在一定的不足，第一点是读数的误差。我们无法准确的确定什么时候开始读取小船滑行的长度，造成结果一定的偏差性。同时，由于我们制作的小船比较多，小船在制作过程中质量的问题很难保证；第三点，我们实验中主要考虑了五个主要因素的影响，其他可能影响小船滑行速度的因素被排除在外，所以可能无法找到最优的结果。 以上就是我们整个实验过程的展示，谢谢大家。

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