37252 Regression Analysis

　　Lab 7: Weighted Least Squares Regression

　　在本周的实验中，我们以不同的方式使用回归，这一次是比较两个观察员对鸡群计数的估计准确性。下表总结了我们现在考虑的变量。

Name	Role	Description
	response	true flock count
	predictor	estimated flock count observer 1
	predictor	estimated flock count observer 2

　　场景。在加拿大哈德逊湾以西的夏季范围内，已使用航测方法估算雪雁的数量。小型飞机在范围内飞行，当发现鹅群时，有经验的观察员会估计鹅群中的鹅数。该方法显然可以用于其他类型的鸟类，在澳大利亚，也可以用于成群的袋鼠或水牛数量。但这是一种可靠的方法吗?一项研究通过在飞机上使用两名独立的观察员对此进行了调查，并用照片备份了他们的观察结果。后来，该照片用于获得鸟类数量的准确计数。获得photo与obs1以及photo与obs2的散点图(可以做矩阵散点图)。如果您对photo vs obs1或photo vs obs2进行了简单的线性回归，您会期望哪些问题[1分]?为什么将照片作为响应变量[1分]是合适的?假设观察员素质高，为什么简单线性回归比使用obs1和obs2作为自变量的多重线性回归更合适[1分]?

　　残留散点图显示了简单线性回归模型[1分]的拟合度有哪些问题?您可以采取什么行动[1分]?由于残差的变异性似乎与自变量成比例地增加，因此我们可以尝试WLS。令(ε1)̂_i代表以obs1为预测因子的简单OLS模型的残差，而(ε2)̂_i代表以obs2为预测因子的模型的残差。假设var((ε1)̂_i)=〖obs1〗_iσ^ 2和var((ε2)̂_i)=〖obs2〗_iσ^ 2。照片与obs1 [1分]的WLS回归的权重是多少?如果我们想直接转换模型[1分]，我们将数据乘以什么?运行WLS回归在运行WLS回归模型之前，我们需要创建权重变量。首先是obs1的权重。使用obs1为SPSS创建权重以用于回归模型。SPSS程序菜单转换>计算变量…在“目标变量：”文本框中单击，然后键入w1。单击数值表达式：文本框，键入1 / obs1点击确定

　　Modify the last procedure to create the weights for the model with .

　　Now fit a WLS regression model with as response, as predictor and as weights.

　　SPSS procedure

　　Copy options in screen shots below

　　Click OK

　　Repeat this procedure to create a model with as response, as predictor and as weights.

　　Computing WLS residuals

　　We now have to calculate residuals suitable for assessing the regression assumptions (the unstandardised residuals are not weighted and are not appropriate for the task).

　　Step 1 – computing the weighted residuals. Using the Compute Variable feature, create the weighted residuals and by multiplying the raw residuals (the unstandardized residuals saved when running the regression) by the square root of the weights.

　　Some descriptive statistics of these weighted residuals are copied below.

　　Step 2 – standardising the weighted residuals. Using the Compute Variable feature, create the standardised, weighted residuals

　　and

　　Some descriptive statistics of these standardised, weighted residuals are copied below.

　　(a) For both WLS models, analyse the Student-T version of the standardised, weighted residuals using scatter plots involving the independent variables. Has weighting improved the behaviour of the residuals [2 marks]?

　　(b)For both WLS models, describe the results of the two-sided T-test with null hypothesis [2 marks]. Explain, in the context of these models, why we are interested in such tests [2 marks].

　　(c) For both WLS models, perform a two-sided T-test with null . Write down the null and alternative hypotheses [2 marks], the test-statistic [2 marks] and the result of the test with reason [2 marks]. Explain, in the context of these models, why we are interested in such tests [2 marks].

　　(d)With reference to your answers in (e) and (f), which observer should be preferred [1 mark]? Why [2 marks]