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Experimental design methods allow these problems to be solved ef ciently during the early stages of the product cycle. This has the potential to dramatically lower overall product cost and reduce development lead time.

EXAMPLE 36-8

14-6 FACTORIAL EXPERIMENTS WITH RANDOM FACTORS (CD ONLY)

(continued)

In this chapter, we focus primarily on the case where all the factors are xed; that is, the experimenter speci cally chose the levels, and the conclusions from the experiment are con ned to

private function onClick(event:MouseEvent):void { bmp.floodFill(event.localX, event.localY, Math.random() * 0xffffff); } } }

14-4

those speci c levels. Now we brie y consider the situation where one or more of the factors in a factorial experiment are random, using the two-factor factorial design as an illustration. The Random-Effects Model Suppose that we have two factors A and B arranged in a factorial experiment in which the levels of both factors are selected at random from larger populations of factor levels, and we wish to extend our conclusion to the entire population of factor levels. The observations are represented by the model 1 2 ij i ij k j k 1, 2, . . . , a 1, 2, . . . , b 1, 2, . . . , n

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Recall from 34 that you can change the appearance of any display object by modifying its color transform. BitmapData is not a DisplayObject, but you can still apply a color transform to it. Unlike display objects, which retain their transformations and are continually affected by them, bitmaps are static, so the colorTransform() method permanently modi es the color of every pixel. The colorTransform() method requires two parameters: a Rectangle object de ning the region to which to apply the color transform, and a ColorTransform object to use. Color transforms are really useful when you want to use a bitmap as a frame buffer, drawing new frames on top of older frames. To create a great motion trail effect with little effort, just dim the previous contents of the frame before drawing a new frame, as shown in Example 36-9. EXAMPLE 36-9

(S14-1)

package { import flash.display.*; import flash.events.*; import flash.geom.*; [SWF(backgroundColor="#000000",frameRate="60")] public class ch36ex9 extends Sprite { protected const MAXSPEED:Number = 8; protected var ball:Shape; protected var holder:Sprite; protected var bmp:BitmapData; protected var velocity:Point; public function ch36ex9() { bmp = new BitmapData(stage.stageWidth, stage.stageHeight); var bitmap:Bitmap = new Bitmap(bmp); addChild(bitmap); holder = new Sprite(); addChild(holder); ball = new Shape(); ball.graphics.lineStyle(0, 0xff0000); ball.graphics.beginFill(0xffffff, 0.1); ball.graphics.drawCircle(0, 0, 30); holder.addChild(ball); stage.addEventListener(Event.ENTER_FRAME, onEnterFrame); stage.addEventListener(MouseEvent.CLICK, onMouseClick); velocity = new Point(1, 1); }

where the parameters i, j, ( )ij, and ijk are normally and independently distributed random variables with means zero and variances 2, 2 , 2 , and 2, respectively. As a result of these assumptions, the variance of any observation Yijk is V 1Yijk 2

protected function onEnterFrame(event:Event):void { ball.x += velocity.x; ball.y += velocity.y; ball.rotation += velocity.length / 5; if (ball.x >= bmp.width || ball.x <= 0) velocity.x *= -1; if (ball.y >= bmp.height || ball.y <= 0) velocity.y *= -1; bmp.colorTransform(bmp.rect, new ColorTransform(1, 1, 1, 0.9)); bmp.draw(holder); } protected function onMouseClick(event:MouseEvent):void { ball.x = stage.mouseX; ball.y = stage.mouseY; velocity.x = Math.random() * MAXSPEED * 2 - MAXSPEED; velocity.y = Math.random() * MAXSPEED * 2 - MAXSPEED; } } }

2 2 2 2

In the example, you implement a simple motion trail by drawing every frame into the same bitmap, while fading out the contents continually with a ColorTransform.

and 2, 2 , 2 , and 2 are called variance components. The hypotheses that we are interested in testing are H0: 2 0, H0: 2 0, and H0: 2 0. Notice the similarity to the single-factor experiment random-effects model discussed in 13. The basic analysis of variance remains unchanged; that is, SSA, SSB, SSAB, SST, and SSE are all calculated as in the xed-effects case. To construct the test statistics, we must examine the expected mean squares. They are E1MSA 2 E1MSB 2 and E1MSE 2

catch (InterruptedException e) System. out .println("Sleepuinterrupted."); 1

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