• What Is An Experiment Quizlet
• Unbalanced Incomplete Block Design

The incomplete block designs can use lesser number of television sets or LCD panels to conduct the test of significance of treatment effects without losing, in general, the efficiency of design of experiment. A valid experiment is the one which is done on the basis of some facts and figures. The experiment which has a good statistical analysis is known to be valid experiment.

Generate Reference Book:may be more up-to-dateTwo level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the 'vital few' factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.2 k DesignsThe factorial experiments, where all combination of the levels of the factors are run, are usually referred to as full factorial experiments. Full factorial two level experiments are also referred to as designs where denotes the number of factors being investigated in the experiment. In Weibull DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below. A full factorial two level design with factors requires runs for a single replicate.

For example, a two level experiment with three factors will require runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from to, then the two levels used in the design for this factor would be and.The two levels of the factor in the design are usually represented as (for the first level) and (for the second level). Note that this representation is reversed from the coding used in for the indicator variables that represent two level factors in ANOVA models.

For ANOVA models, the first level of the factor was represented using a value of for the indicator variable, while the second level was represented using a value of. For details on the notation used for two level experiments refer to.The 2 2 DesignThe simplest of the two level factorial experiments is the design where two factors (say factor and factor ) are investigated at two levels.

A single replicate of this design will require four runs ( ) The effects investigated by this design are the two main effects, and and the interaction effect. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by; represents the treatment combination where factor is at the high level or the level of, while the remaining factors (in this case, factor ) are at the low level or the level of.

Similarly, represents the treatment combination where factor is at the high level or the level of, while factor is at the low level and represents the treatment combination where factors and are at the high level or the level of the 1. Figure (b) below shows the design matrix for the design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero.

As a result the design is an orthogonal design. In fact, all designs are orthogonal designs. This property of the designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.The design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below. The 2 3 DesignThe design is a two level factorial experiment design with three factors (say factors, and ). This design tests three ( ) main effects, and; three ( ) two factor interaction effects,; and one ( ) three factor interaction effect,.

The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are,. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the standard order or Yates' order.

The design is shown in figure (a) below. The design matrix for the design is shown in figure (b).

The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns. The design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.Analysis of 2 k DesignsThe designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large.

The use of designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for designs because all factors are only at two levels. The applicable model using the notation for designs is:where the indicator variable, represents factor (honing pressure), represents the low level of 200 and represents the high level of 400. Similarly, and represent factors (number of strokes) and (cycle time), respectively. Is the overall mean, while, and are the effect coefficients for the main effects of factors, and, respectively., and are the effect coefficients for the, and interactions, while represents the interaction.If the subscripts for the run (; 1 to 8) and replicates (; 1,2) are included, then the model can be written as:To investigate how the given factors affect the response, the following hypothesis tests need to be carried:This test investigates the main effect of factor (honing pressure). The statistic for this test is:where is the mean square for factor and is the error mean square.

Hypotheses for the other main effects, and, can be written in a similar manner.This test investigates the two factor interaction. The statistic for this test is:where is the mean square for the interaction and is the error mean square.

Hypotheses for the other two factor interactions, and, can be written in a similar manner.This test investigates the three factor interaction. The statistic for this test is:where is the mean square for the interaction and is the error mean square.To calculate the test statistics, it is convenient to express the ANOVA model in the form.Expression of the ANOVA Model asIn matrix notation, the ANOVA model can be expressed as:where:Calculation of the Extra Sum of Squares for the FactorsKnowing the matrices, and, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical.

The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor is:where is the hat matrix and is the matrix of ones. The matrix can be calculated using where is the design matrix, excluding the second column that represents the main effect of factor. Thus, the sum of squares for the main effect of factor is:Similarly, the extra sum of squares for the interaction effect is:The extra sum of squares for other effects can be obtained in a similar manner.Calculation of the Test StatisticsKnowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction is:where is the mean square for the interaction and is the error mean square. The value corresponding to the statistic, based on the distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:Assuming that the desired significance is 0.1, since value 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums.

Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to. The coefficients and related results are shown in the Regression Information table above.

In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error,. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the statistic, corresponding to the coefficients.

The P Value column displays the value corresponding to the statistic. (For details on how these results are calculated, refer to ). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.Model EquationFrom the analysis results in the above figure within section, it is seen that effects, and are significant. In a DOE folio, the values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present design in terms of the coded values can be written as:To make the model hierarchical, the main effect, needs to be included in the model (because the interaction is included in the model).

The resulting model is:This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon. Replicated and Repeated RunsIn the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels.

However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs.

If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b). Unreplicated 2 k DesignsIf a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.However, sometimes it is only possible to run a single replicate of the design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated designs are available.

These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.Pooling Higher Order InteractionsOne of the ways to deal with unreplicated designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square.

Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.Normal Probability Plot of EffectsAnother way to use unreplicated designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in, the standard error for all effect coefficients in the designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with ) will fall along the straight line representative of the normal distribution, N( ). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated designs, remains unknown since cannot be obtained.

Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line.

Themethod is illustrated in the following example.ExampleVinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor ), material temperature (factor ), element intensity (factor ) and pre-stretch (factor ). The two levels of the factors used in the experiment are as shown in below. With a design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.The experiment design and data, collected as percent defects, are shown in the following figure.

Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares,. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure. Lenth's method uses these values to estimate the variance. As described in, if all effects are arranged in ascending order, using their absolute values, then is defined as 1.5 times the median value:Using, the 'pseudo standard error' ( ) is calculated as 1.5 times the median value of all effects that are less than 2.5:Using as an estimate of the effect variance, the effect variance is 2.25.

Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects, and the interaction do not follow the distribution represented by this line. Therefore, these effects are significant.The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure).

If the required significance is 0.1, then:The statistic, is calculated at a significance of (for the two-sided hypothesis) and degrees of freedom number of effects. Thus:The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant.

These effects are, and the interaction. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels. Center Point ReplicatesAnother method of dealing with unreplicated designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error.

Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in.Example: Use Center Point to Get Pure ErrorConsider a experiment design to investigate the effect of two factors, and, on a certain response. The energy consumed when the treatments of the design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure.

The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next. Since the present design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, can be used to obtain an estimate of pure error,. Let represent the average response for the five replicates at the center. Then:Then the corresponding mean square is:Alternatively, can be directly obtained by calculating the variance of the response values at the center points:Once is known, it can be used as the error mean square, to carry out the test of significance for each effect.

For example, to test the significance of the main effect of factor the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original design.Then, the test statistic to test the significance of the main effect of factor is:The value corresponding to the statistic, based on the distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:Assuming that the desired significance is 0.1, since value. Using Center Point Replicates to Test CurvatureCenter point replicates can also be used to check for curvature in replicated or unreplicated designs. The test for curvature investigates whether the model between the response and the factors is linear. The way the DOE folio handles center point replicates is similar to its handling of blocks.

The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.Example: Use Center Point to Test CurvatureTo illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate experiment from a preceding figure(labeled ' 2 2 design augmented by five center point runs'). Let be the indicator variable to indicate if the run is a center point:If and are the indicator variables representing factors and, respectively, then the model for this experiment is:To investigate the presence of curvature, the following hypotheses need to be tested:The test statistic to be used for this test is:where is the mean square for Curvature and is the error mean square.Calculation of the Sum of SquaresThe matrix and vector for this experiment are:The sum of squares can now be calculated. For example, the error sum of squares is:where is the identity matrix and is the hat matrix. It can be seen that this is equal to (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the. The number of degrees of freedom associated with, is four.

The extra sum of squares corresponding to the center point replicates (or Curvature) is:where is the hat matrix and is the matrix of ones. The matrix can be calculated using where is the design matrix, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:Calculation of the Test StatisticKnowing the mean squares, the statistic to check the significance of curvature can be calculated.The value corresponding to the statistic, based on the distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:Assuming that the desired significance is 0.1, since value 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above.

The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below. Blocking in 2 k DesignsBlocking can be used in the designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days.

Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the design assume that treatments and were run on the first day and treatments and were run on the second day. Then, the incomplete block design for this experiment is:For this design the block effect may be calculated as:The interaction effect is:The two equations given above show that, in this design, the interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same.

In other words, the interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant.

In this case, it would better to use incomplete block designs that confound these effects with the blocks.One way to design incomplete block designs is to use defining contrasts as shown next:where the s are the exponents for the factors in the effect that is to be confounded with the block effect and the s are values based on the level of the the factor (in a treatment that is to be allocated to a block). For designs the s are either 0 or 1 and the s have a value of 0 for the low level of the th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the design where the interaction effect is confounded with the block. Since there are two factors, with representing factor and representing factor.

Therefore:The value of is one because the exponent of factor in the confounded interaction is one. Similarly, the value of is one because the exponent of factor in the confounded interaction is also one. Therefore, the defining contrast for this design can be written as:Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the design, there are four treatments,. Assume that represents block 2 and represents block 1. In order to decide which block the treatment belongs to, the levels of factors and for this run are used. Since factor is at the low level in this treatment,.

Similarly, since factor is also at the low level in this treatment,. Therefore:Note that the value of used to decide the block allocation is 'mod 2' of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of, treatment is assigned to block 1. Other treatments can be assigned using the following calculations:Therefore, to confound the interaction with the block effect in the incomplete block design, treatments and (with ) should be assigned to block 2 and treatment combinations and (with ) should be assigned to block 1.Example: Two Level Factorial Design with Two BlocksThis example illustrates how treatments can be allocated to two blocks for an unreplicated design. Consider the unreplicated design to investigate the four factors affecting the defects in automobile vinyl panels discussed in.

Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction was not significant and decided to allocate treatments to the two operators so that the interaction was confounded with the block effect (the two operators are the blocks).

What Is An Experiment Quizlet

The allocation scheme to assign treatments to the two operators can be obtained as follows.The defining contrast for the design where the interaction is confounded with the blocks is:The treatments can be allocated to the two operators using the values of the defining contrast. Assume that represents block 2 and represents block 1. Then the value of the defining contrast for treatment is:Therefore, treatment should be assigned to Block 1 or the first operator. Similarly, for treatment we have.

Therefore, should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.In a DOE folio, to confound the interaction for the design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull DOE folio is shown in the third of the following figures.

This design matches the allocation scheme of the preceding figure. For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the interaction is considered as the sum of squares due to blocks. In the DOE folio, this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed.

Since the design is unreplicated, any of.

Incomplete Dominance & Mendel's Experiment“Incomplete dominance is a form of intermediate inheritance in which one allele for a particular trait is not expressed completely over its paired allele.”Incomplete dominance is a form of Gene interaction in which both alleles of a gene at a locus are partially expressed, often resulting in an intermediate or different phenotype. It is also known as partial dominance.For eg., in roses, the allele for red colour is dominant over the allele for white colour. But, the heterozygous flowers with both the alleles are pink in colour. Mechanism of Incomplete DominanceIncomplete dominance occurs because neither of the two alleles is completely dominant over the other. This results in a phenotype that is a combination of both.Gregor Mendel conducted experiments on pea plants.

He studied on seven characters with contrasting traits and all of them showed a similar pattern of inheritance. Based on this, he generalized the law of inheritance.Later, researchers repeated Mendel’s experiment on other plants. Shockingly, they noted that the F1 Generation showed variation from the usual pattern of inheritance.

Unbalanced Incomplete Block Design

The monohybrid cross resulted in F1 Progeny which didn’t show any resemblance to either of the parents, but an intermediate progeny.Let’s understand the incomplete dominance with the example of Snapdragon ( Antirrhinum sp).Monohybrid cross was done between the red and white coloured flowers of Snapdragon plant. Consider, pure breed of the red flower has RR pair of alleles and that for the white flower is rr.Firstly, true breeding red (RR) and white (rr) coloured flowers of snapdragon were crossed.

The F1­ generation produced a pink coloured flower with Rr pair of alleles.Then the F1 progeny was self-pollinated. This resulted in red (RR), pink (Rr) and white (rr) flowers in the ratio of 1:2:1.Recollect that the genotype ratio of F2 generation in the monohybrid cross by Mendel also gave the same ratio of 1:2:1. However, the phenotype ratio has changed from 3:1 to 1:2:1. The reason for this variation is the incomplete dominance of the allele R over the allele r. This led to the blending of colour in flowers.Also Read: Concept of DominanceIn genetics, Dominance is a relationship between alleles of one gene.

In order to understand the concept of the dominance of alleles, we need to know more about.So far we know that genes are a hereditary unit in organisms which exist as a pair of alleles in diploid organisms. These pair of alleles may or may not be similar. That is, a heterozygous gene has two dissimilar pairs of alleles while homozygous have identical ones.Heterozygous alleles carry different information on traits. When we say one trait is dominant over the other, there can be two reasons:.

either it is non-functional, or. is less active than the normal alleleIncomplete Dominance and CodominanceIncomplete dominance and codominance are different from each other.In codominance, both the alleles present on a gene are expressed in the phenotype. A flower showing codominance will have patches of red and white instead of a uniformly pink flower.In incomplete dominance, the F2 generation from heterozygous plants will have a ratio of 1:2:1 with the phenotypes red, white and spotted flowers.The humans with AB blood type also show codominance where the alleles for both blood types A and B are expressed. Examples of Incomplete DominanceExamples of incomplete dominance are mentioned below: In HumansThe child of parents each with curly hair and straight hair will always have wavy hair. Carriers of Tay-Sachs disease exhibit incomplete dominance. In Other AnimalsThe Andalusian chicken shows incomplete dominance in its feather colour.When the rabbits with long and short furs are bred, the offsprings produced will have medium fur length.Also Read:For more details on incomplete dominance, dominance and incomplete dominance and codominance, keep visiting BYJU’S website or download BYJU’S app for further reference.