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This general procedure can be extended to confounding the 2k design in 2p blocks, where p k. Start by selecting p effects to be confounded, such that no effect chosen is a generalized interaction of the others. Then the blocks can be constructed from the p de ning contrasts L1, L2, . . . , Lp that are associated with these effects. In addition to the p effects chosen to be confounded, exactly 2p p 1 additional effects are confounded with blocks; these are the generalized interactions of the original p effects chosen. Care should be taken so as not to confound effects of potential interest. For more information on confounding in the 2k factorial design, refer to Montgomery (2001, 7). This book contains guidelines for selecting factors to confound with blocks so that main effects and low-order interactions are not confounded. In particular, the book contains a table of suggested confounding schemes for designs with up to seven factors and a range of block sizes, some of which are as small as two runs. EXERCISES FOR SECTION 14-8

You can load the le into a ByteArray, decode it, write the bitmap to a BitmapData, and display it in a Bitmap Do that, and you ve just eclipsed the capabilities of Flash Player! This example addresses another important thing about bitmaps They are always stored uncompressed It s essential to quickly have access to any pixel value, a property that storing the data compressed wouldn t allow for This means that the data stored in image les like PNGs and JPGs are not bitmaps but they can be decompressed and the bitmap data can be extracted As an alternative to the display list, bitmaps are great for games! Tile engines use small square textures tiled across the screen to easily generate huge terrains, usually for top-down adventure games or strategy games.

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14-23. Consider the data from the rst replicate of Exercise 14-13. Suppose that these observations could not all be run under the same conditions. Set up a design to run these observations in two blocks of four observations each, with ABC confounded. Analyze the data. 14-24. Consider the data from the rst replicate of Exercise 14-14. Construct a design with two blocks of eight observations each, with ABCD confounded. Analyze the data. 14-25. Repeat Exercise 14-24 assuming that four blocks are required. Confound ABD and ABC (and consequently CD) with blocks. 14-26. Construct a 25 design in two blocks. Select the ABCDE interaction to be confounded with blocks. 14-27. Construct a 25 design in four blocks. Select the appropriate effects to confound so that the highest possible interactions are confounded with blocks. 14-28. Consider the data from Exercise 14-18. Construct the design that would have been used to run this experiment in two blocks of eight runs each. Analyze the data and draw conclusions. 14-29. An article in Industrial and Engineering Chemistry ( Factorial Experiments in Pilot Plant Studies, 1951, pp. 1300 1306) reports on an experiment to investigate the effect of temperature (A), gas throughput (B), and concentration (C) on the strength of product solution in a recirculation unit. Two blocks were used with ABC confounded, and the experiment was replicated twice. The data are as follows:

Texture atlases use one large bitmap to store different frames of animation; moving the area that s drawn from the atlas to the screen provides a computationally cheap way to animate prerendered game sprites A frame buffer stores a bitmap as it s built, eventually drawing to the screen at once You can use frame buffers to incorporate elements of the last frame into the current frame, for example, to draw trails, smoke, re, halos, and so on All of these old-school game tricks are easily implemented with bitmaps One application of bitmaps is important for more than just games You can use bitmaps to cheaply scale up content to great sizes For example, drawing a complex scene using the display list at 1600 1200 can really hurt your frame rate, especially if you re using lters (covered in 37, Applying Filters ).

Replicate 1 Block 1 (1) ab ac bc 99 52 42 95 Replicate 2 Block 3 (1) 46 ab 47 ac 22 bc 67 Block 4 a 18 b 62 c 104 abc 36 Block 2 a b c abc 18 51 108 35

Instead, you can draw the scene at 640 480, render it to a bitmap, and scale the bitmap up 25 The end result will lose some clarity by being scaled up, but the frame rate is sure to improve drastically You don t need to use this trick for video or if you ve already chosen to use full-screen with a fullScreenSourceRect; both use hardware acceleration in the latest versions of Flash Player..

(a) Analyze the data from this experiment. (b) Analyze the residuals and comment on model adequacy. (c) Comment on the ef ciency of this design. Note that we have replicated the experiment twice, yet we have no information on the ABC interaction.

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