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separately Finally, propensity scoring does not require parametric assumptions about outcome distributions Propensity scoring can be viewed as a natural extension of survey weighting With two groups a treatment group and a control group the analyst first develops a logistic (or other) model to predict group membership based on the potential confounders One approach uses inverses of the predicted propensity scores as weights in the analysis An alternative approach, which reduces variance while eliminating slightly less bias, is to divide the sample into a small number of strata (say five) based on the propensity scores The groups are then reweighted so that the weighted count in each stratum is the same in the treatment and control groups The reweighting thus approximately equates the groups in terms of the confounders.

10 3

If each group is weighted to conform to the overall estimated population distribution, the estimate of the outcome for each group can be viewed as an estimate of the outcome had the whole population been in that group Using an ordinal logit model, the propensity scoring methodology can be extended to cover several treatment groups (Joffe and Rosenbaum, 1990) When the same outcome measure is collected at multiple points in time, the question of whether to use an endpoint or a change score analysis arises The dependent variable in an endpoint analysis is typically the latest measurement of the outcome, whereas the dependent variable in a change score analysis is the difference between measurements of the outcome at two waves.

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An endpoint analysis often includes the baseline measure as one of the confounders, and the change score analysis often defines the change score as the difference between the latest outcome measurement and the baseline (or earliest) measurement When an outcome is measured on an ordinal scale with only a few response categories, like the Likert items in the YMCLS, a change score analysis can be problematic because of floor and ceiling limitations An endpoint analysis without control on the baseline outcome measure is more powerful than a change score analysis if the correlation of the measured outcomes at the two time points is less than 05 However, there are also bias considerations If the baseline outcome measure is both correlated with treatment and subject to measurement error, then an endpoint analysis with control on the baseline outcome can yield biased effect estimates (Cook and Campbell, 1979, pp.

160 164), although the bias may be reduced if multiple baseline measurements are used (Cook and Campbell, 1979, pp 170 175) Analysing change scores without controlling for baseline outcomes can remove this bias under some but not all conditions (Allison, 1990; Bonate, 2000; Senn, 2006) 1725 Value of Longitudinal Surveys.

expensive. Combined with reactive ion etching (RIE), creation of DOEs with very ne resolution and high ef ciency is possible. As an example, consider a binary phase DOE consisting of a 2-D array of small apertures that can be assumed to be sampled points. At each such point, a phase shift of either 0 or 180 degrees is to be generated. At points where a 180 degree phase shift is desired, the substrate is etched to the required depth. The e-beam exposes a pattern onto an e-beam sensitive resist indicating where apertures are to be made, and RIE is used to etch the apertures to the desired depth. This can also be achieved by chemical processing, but RIE is a higher precision technology. This approach can be generalized to any number of phase quantization levels in powers of 2. A typical e-beam machine is capable of generating a beam with a spot size around 10 nm. Therefore, it is capable of exposing patterns with lines of widths less than 0.1 mm or 100 nm. For binary phase DOEs, each square aperture has a size of approximately 1 mm 1 mm. For such feature sizes around, a total pattern size of, say, 1 mm 1 mm can be achieved, while maintaining high resolution. Larger patterns are generated by stitching together a number of such patterns together under interferometric control. Dry etching systems are capable of generating very ne surface relief. For example, 180 degree phase change needed for binary phase DOEs occurs over a distance of half a wavelength. For visible light, that corresponds to a length of the

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