Theorem: Covariance Among Treatment Pairs in Randomized Block Designs Yields Coefficient of Covariance for Randomized Blocks

The only true and complete Randomized Block Theorem for Statistical Designs ---------> Unified Field Home Page
© Copyright Alan R. Foos, Jan-Aug 2007

Problem and Postulate.

For any set of randomized blocks in a uniform gradient, there exists a number r such that r varies from zero to one as the gradient varies from parallel to perpendicular. This can be called the coefficient of covariance for randomized blocks.

We start with the observation that for variation among replications in a randomized block ANOVA design to be significant, it must incorporate positive covariance among the replicated treatments (covariance of all treatment pairs from replication to replication). Mathematical derivations will be used to define exactly what the relationship between covariance and replication significance is and understand how block to block treatment variation is useful in reducing sum of squares and increasing experimental precision. An r value is calculated that represents the fraction of covariance for all combinations of treatment pairs across blocks to the total block to block variance.

Ideally, assuming the direction of one variable gradient or vector sum of several across blocks is uniform, the r value of treatment pairs would indicate the deviation from right angles that it intersects replications, the r value (not the same as the conventional correlation coefficient) of 1 corresponding to a perpendicular intersection, and an r value of zero meaning an intersection of zero angle (blocks are parallel to the gradient). Such information could then be used to increase precision in subsequent experimental designs. If the extraneous variable can be identified and measured, the magnitude and direction of the gradient can also be estimated and used with other regression variables in the experiment for predicting outcomes with better precision. It is important for the reader to understand that this theorem represents a fundamental and original insight into the established principles of statistical design, so that any reliance on recent developments in the field is neither necessary nor relevant. Supporting citations are dated according to the approximate time that the theorem was first developed, and the scientific principals have not been otherwise subject to change.

CAUTION: If a set of parallel blocks is intersected by any uniform gradient, or subsequently rotated as a unit around a common center, an oblique angle will be measured as a perpendicular intersection. It is the lack of uniformity from block to block that produces covariance and r values. If there is no error term, i.e., some background noise, then any uniformly measurable differences in blocks produced by the same angle of incidence other than zero will produce r values of one because all cells may be affected to a consistent degree, thus masking distortion in treatment effects. To avoid distortion among treatment effects it is necessary to ensure that the primary vector intersection is perpendicular to all blocks. Not only will an oblique angle cause distortion in treatment effects, but it will reduce the block mean square and mask block differences (BMS). It should also be stressed that any lack of uniformity among cells from block to block, and only such, will produce r values less than one (and will increase EMS). The section on the Equivalence Principle and Gradient wheel (Gradius) discuss how to identify a primary gradient.

The "candy bar" (representing a set of blocks lengthwise) near the end of this paper may also help visualize this concept using a lighting gradient. The strongest light is towards the upper right. It is not parallel because it is strongest between top and bottom, not side to side. But it is not quite perpendicular, either, because it also varies from side to side. Not only that, but it is not quite uniform, because it is strongest along a clearly visible axis of origin. As a result, the optimum value of r will be less than one for any set of blocks, but there is clearly a definite origin to the light source, so that there is an r value less than one which represents optimum orientation. Let us assume that optimum value for r is 0.8, meaning that we can expect some distortion in any experimental results because of the lack of gradient uniformity, but we can minimize that by optimum orientation (perpendicular).

If we were to calculate the precise r value, it would be somewhat less than the optimum of 0.8, perhaps 0.7, because the light source does not intersect the top of the block in the exact center. The only way to know the optimum r value would be to measure the gradient before the experiment is conducted, precisely at points across a set of blocks, and inspect the results to see if the balance from end to end of the blocks is uneven (average treatment variances across blocks are not equal even though covariance is more than zero). If so, then there is clearly an origin to the gradient and blocks can be adjusted to achieve optimum balance. Since there is an obvious origin (light source) to the gradient, the optimum r value will be also be slightly less. If the existence of an intersecting variable is not known, the value and direction of r indicating consistent block to block effects could reveal its presence, degree of or lack of uniformity and suggest improvement in block orientation. The use of two sets of blocks arranged at varying angles could also be helpful in verifying the direction, origin and magnitude of a gradient variable.

It may also be helpful to view the following chart which represents the plot layout in my 1976-1979 thesis work. Each pink or blue strip is a block of treatments. Part of the experiment was also a comparison between two forages, tthe pink strip is sainfoin and the blue is alfalfa. The treatments for both were identical and included two factorial sets for N, P and K fertilizer treatments and several micronutrient paired comparisons. The cell numbers in the upper right corners refer to the treatment number after randomizing. The chart just below that is how the matching field plot appeared after planting. My concerns over gradient effects first led to the development of the randomized block theorem in 1978, but the equations were difficult to present without computer assistance. Several versions were written after that until the last in 2007.

Section 2. Introduction.

The proposal is based on the fact that if a gradient (or trend) exists among replications in a randomized block design (the assumption usually made for randomized block experiments), that all treatments are affected equally (i.e, exposed equally in each separate block) from one replication to another. This is the ideal case only, in which each set of replications must be oriented at the same right angle to the gradient. This ideal is seldom realized in practice, but provides opportunity in certain cases for measuring extraneous variables of interest. This trend can be demonstrated as covariance among treatments, and an r value (1 to zero) calculated to estimate the degree of efficiency in terms of the angle of gradient intersection (90 degrees to zero). Two ANOVA models are in common practice. The first model is what is called a CR (completely randomized) and the second is called an RB (randomized block), where TMTs (treatments) are randomized within each replication. Each replication then comprises one member of a set of randomized blocks. This proof provides a method for measuring gradients, optimizing design layout, and measuring block orientation.

In a CR design, the more replications are used, the more precision is obtained as reflected in smaller ESS (error sums of squares). This is not directly measurable since replication variance does not enter into the ANOVA table; thus, completely randomized replications do not increase experimental precision under controlled circumstances designed to directly filter it. The CR is also called a one-way design. In the RB, or two way, design, the overall SS (sum of squares) is reduced by both treatment sum of squares (TSS) and block sum of squares (BSS). The ratio of mean squares obtained by dividing by degrees of freedom (number of treatments or blocks less one) is called F, and corresponds to the probability that the differences are caused by chance (the null hypothesis). The required probability for rejecting the null hypothesis, p(F), may be read from a table or an exact p(F) calculated for the F value. Precision is not always gained by putting replications into discrete blocks - it can even be reduced; hence, there may be no point in blocking.

How can that be when variation is always positive? For several reasons, one being that the amount of variation is a measure of significance only when divided by remaining degrees of freedom, so the TMT F ratio may actually be decreased by introducing blocks that reduce both degrees of freedom and error SS. In fact, that must be the case unless there is an extraneous variable intersecting blocks that causes true differences among them (this implies TMT covariance from block to block). The variable may be known or unknown, and the gradient may or may not (as in agricultural field plots) be perfectly uniform. Another problem is that even if TMT precision is gained, the results can be erroneous if the extraneous variable is not independent of TMT variation. This can happen even if the variable itself would not interact normally, but does so anyway because the gradient intersects a small number of blocks at less than 90 but more than 0 degrees, thus affecting some treatments more than others.

Even if the interaction is zero, to the extent to which blocks are not arranged perpendicular to the extraneous or intersecting variable, any reduction in the SS will be lost because blocks will be equally affected and will thus not reduce the EMS (error mean square). Snedecor and Cochran, page 256, point out by means of an example; “Note that the replications (block) mean square is more than twice the residuals mean square, indicating real differences between replication means and suggesting that the classification into blocks was successful in improving accuracy.” This implies that if no real differences among replications (as blocks) exist, then there was no point in going to the trouble of blocking, even though there may be plenty of reasons to have that many replications in a CR design; in fact, more random replications will be required to match any precision gained by using blocks.

Mendenhall, page 653, states “blocks may represent time, location, or experimental material. Thus, if three TMTs are to be compared and there is a suspected trend, a substantial part of the variation can be removed by blocking.” It is fair to restate this and say that if there is no block to block gradient or trend, then no variation can be removed. The question may arise as to how much variation is necessary among blocks for there to be success (for a reduction in EMS). This may not be initially easy to grasp. Mendenhall provides a section called Some Cautionary Comments on Blocking, where these points are covered on page 712: “... you should not use a block design to investigate the effects of two factors... if the two factors do not affect the response independently of one another (i.e., zero interaction), then a randomized block ANOVA could lead to very erroneous conclusions.” Thus, a moisture gradient as is often the object in agricultural designs, should (ideally) cause uniform changes in yield for all treatments from block to block (covariance) since otherwise the block - treatment interactions will distort results whether both factors are considered or not. Because such interactions are usually ignored and block orientation not analyzed, this represents a serious and unnecesssary limitation in RB designs. If regression analaysis is used to incorporate the block variable, appropriate interactions with all treatment variables will be identified with mathematical precision and filtered from the error term as with any other treatment factorials. Of course, only RB designs with measurable gradients will permit this.

Hence, the question arises as to whether ignoring the nature and effect of a gradient is the most responsible course in certain cases. For many purposes, the effect of a gradient represents uncontrolled conditions which may not be representative of those found in the field, and failure to identify and measure it brings the applicability of results into question, even more so if gradient interaction with treatments exists. For example, if a moisture gradient effects yields, the ideal would be to identify and measure the gradient as well as its effects. If moisture does in fact cause an interaction, then treatment effects will be distorted whether the effects of a block factor are ignored or not. Alternatively, it is possible in some cases to incorporate the moisture variable as a potential factor in the experimental design and statistically determine the effect of the gradient variable including interactions with treatments. This can be performed with potentially high precision using standard regression techniques.

In reality, an assumption of non interaction may not be strictly true, but it should be close even if the effect of an extraneous variable is ignored since such distortions will also affect intrinsic variables of interest. Therefore, the RB design is usually used to filter (or average) the effects of nuisance, extraneous variables. These are effects that cannot be precisely controlled or measured and which may cause treatment interactions which are less severe if filtered, and it is reassuring when blocks are significant. It is, however, not always the case that extraneous variables cannot be measured, that these are not sufficiently uniform, or that their interaction with treatments is of any more concern for the extraneous variable than their unseen effect on the treatments under consideration. Additionally, a carefully constructed RB design can be used to great advantage for measuring effects of extraneous variables if experimental controls are sufficient and blocks consistently oriented.

Mendenhall continues... “The second point is that blocking is not always beneficial... blocking produces a gain in information (ONLY) if the between-block variation (due to covariance effects) is larger than the within-block variation (more precisely, as will be seen, the non-covariant or random variation for treatments between blocks). Then blocking removes this larger source of variation from SSE (sum of squares for error) and EMS (error mean square) assumes a smaller value because of the design. At the same time, you lose information because blocking reduces the number of degrees of freedom associated with SSE... if blocking is to be beneficial, the gain in information due to the elimination of block variation must outweigh the loss due to a reduction in the number of degrees of freedom associated with SSE.” It is of importance to note that ONLY to the extent that replications are consistently oriented at right angles to the extraneous variable being filtered can there be a significant block variation.

Snedecor and Cochran on page 264-5 define the efficiency of blocking as the fraction of sigma squared for a CR design divided by siqma squared for the corresponding RB. The ratio must exceed the value of one for blocking to succeed, and if the experiment was an RB, then the numbers can be inserted to give the comparative value. They provide an example on that page. Substituting sigma squared with ESS/EDF where EDF is error degrees of freedom, and manipulating terms yields the fact that for blocking to succeed, the ratio of SSE for RB to the SSE for CR must be LESS than the ratio of EDF for RB to EDF for CR. This paper introduces a new measure of efficiency in terms of a combined r value for treatment pairs, the ideal r value being 1 for a perpendicular intersection of blocks in a uniform gradient. An r value of less than one suggests the possibility of improving the design by reorienting blocks.

Note that the EDF for the RB is always less than EDF for the CR, since it is equal to the EDF for the CR design less the degrees of freedom (J-1) for blocks. This means that if blocking is to increase precision, the ratio of the ESS for RB to ESS for CR must be less than a value which is less than one; therefore, the reduction in sums of squares by blocking may very possibly decrease experimental precision. This happens where no real (significant) trend (gradient) exists from block to block and the SS for blocks is not large enough to overtake the loss resulting from lesser degrees of freedom. The extent that it is large enough is related to covariation among treatments across blocks, as a zero value for covariation always produces an F ratio equal to one. By substituting the required F ratio for such an experiment into the equation relating F and covariance, the minimum covariance required for any F value (a least significant covariance (LSCOV)) may be derived. Therefore, blocking is meaningless unless a gradient exists due to an extraneous variable or the vector sum resulting from a combination of variables. Covariance and block significance mutually depend on such an extraneous variable exerting an influence on blocks.

A mathematical equivalent for F in terms of covariance is intuitively implied since blocks cannot differ significantly without covariance among treatments pairs. This way of approaching the issue may be more meaningful than the ratio of two sigmas, since mean square values can be used to calculate covariance effects and r values ranging from zero to one are more informative than F values without upper limits. A trend initially exposed by covariance or calculated r values between two sets of data suggests an extraneous variable is affecting results, and randomized block designs that exploit covariance in such a way can be used to advantage. Since the extraneous variable is not always known (or is a combination), r values for treatment pairs in preliminary experiments can be helpful in isolating and eventually identifying unknowns the importance of which were previously overlooked. Results can also help in designing subsequent experiments. Snedecor and Cochran, p255, observe, “When planning a controlled experiment, the experimenter often acquires the ability to predict roughly the behavior of the experimental material.” The preliminary use of 2-3 treatment replications set at different angles at random, temporally or spatially, can expose hidden variables and suggest larger RB designs that exploit them for increasing experimental precision. Covariance r values may also, in certain cases, have more practical value than treatment variables in the identification of cause and effect relationships; for example, the etiology of diseases and the identification of pre-existing causal links between disease syndromes.

The standard ANOVA for any RB is a shortcut to calculating the combined sum of covariances across blocks. From that, an average covariance or r value for any RB can be readily obtained. The number of degrees of freedom for a randomized block r value depends on the number of treatment pairs in the numerator; that is, the combination of treatments taken two at a time as well as the denominator of t treatments. A function p(r) for the null hypothesis being true should be derivable since p(r=1) would always be zero percent, and p(r=0) would always be 100 percent. In any case, note that p(r) should also be a function of sine theta, where theta is the angle of incidence to randomized blocks by an intersecting gradient. The equations for block mean square, (BMS), error mean square, (EMS), F, average covariance and r for covariance ratio, will be proved:

Section 3. Discussion of Proof. (Refer often to the following figures which illustrate the relationship between F values and treatment covariance across blocks)

F-All Treatments Vary Across Blocks. Note that both treatment and block F values are very large and significant differences would be at a very highly significant level, p(F) <0.001. Covariance exists among treatment pairs.

F-One Treatment Varies Across Blocks. Note that treatment F is greater than one and potentially signicant, while block F is exactly one and never significant, p(F)=1 (100%).

F-Opposing Trends Across Blocks. Note that treatment effects cancel giving an F value of one, and for blocks an F value of zero indicates complete interaction of treatment values with blocks. The case of negative covariance is not suitable for RB design. In such a case, the ratio of average covariance to variance for t treatments will be a negative one.

These relationships will be set in mathematically precise terms and proved in the following sections of this paper. Significance for a randomized block design is judged by F, the ratio of variation in a set of observations, the mean square, to the residual variation, the error mean square.  An F ratio of only one indicates that the numerator variation is the same as the error term, so no real difference exists, associated probability, p(F), then being 100% that there is no real difference among means (the initial null hypothesis being that there is no difference).  Any differences among treatment or replication means would be no greater than that predicted by random variation.  For F ratios greater than 1 there is increasingly less than a 100% chance that means differ by chance alone, because the differences are larger than overall variation.  

The theorem states that any such differences among replications (an F ratio larger than one) would be accountable only by covariance among treatments from one block to the next. Take the simplest case of two treatments.  If treatments in the second replication are consistently greater than in the first, then one should find covariance among  treatments associated with a replication F ratio greater than one. For treatments more than two, the covariance is the sum of all possible combinations among them. Perfectly uniform covariance produces F ratios approaching infinity for both replications and treatments. This is because all variation occurs due to these effects, and error mean squares progressively diminished by block differences will approach zero.  The proof derives a formula for BMS (block mean square) in such terms by breaking out the covariance component to show that the F value for BMS is larger than one only for the sum of individual treatment variances across replications that exceed zero.  The error term thus excludes covariance effects of treatment pairs from block to block.

Interesting deductions follow from this idea. Random, i.e. inconsistent, variation among replications will produce treatment covariances of zero. Random variation will therefore remain in the error term, so F for replications will be one, the largest F ratio permitted given randomized block assumptions. Obviously, because of the need to combine covariances for all treatment combinations, negative covariance is not workable (this is equivalent to treatment interaction with blocks). This leads to logical errors. An example of negative covariance would be an instance where additions of fertilizer increase yields for some treatments in one replication but decrease them for the next. This is a possibility, but it would violate the assumptions for blocks in an RB design. Good practice foresees such effects and designs a factorial experiment that avoids these kind of trends. Perfectly opposing curvatures across replications, meaning perfectly negative covariance among two treatments, results in replication F ratios of zero and treatment F ratios of one; therefore, such interactions are not appropriate for good experimental results.   

In an example, if moisture caused half of the treatments to increase and the other half to decrease, all the variation would be attributed to a significant factorial effect between moisture and fertilizer, and no block differences would result.  If the moisture gradient intersected blocks at a zero angle such that only the ends of each block were affected, then the resulting block to block variation would also be zero, and no block efficiency would result (but, possibly, distortion in treatment effects could result). Thirdly, if only random variation occurs between blocks (BMS=EMS or no gradient exists), then again no reduction in the EMS occurs (except by averaging distortion by using repeated random replications), and the practice of blocking is of no value.

The following proof follows standard induction criteria for k treatments =2 and k + 1 treatments = 3.  In order to make the concept work, the notation for individual treatments across blocks is retained by giving each treatment a separate designation of x, y, z... rather than the subscript of t ordinarily used to reference treatment values (and where j indicates blocks). Two and then three treatments are used for the proof according to standard induction criteria, the permutation of larger numbers being difficult to work with. By using these designations, mean square formulas can be rearranged  to produce individual treatment variances isolated across instead of within replications for rearranging covariance formulas in pairs between them. The result of each case is that replication mean square can be written in terms of the sum of all possible treatment covariance terms and sum of individual treatment variances throughout replications, the ratio for which varies between zero and one as the orientation of blocks changes from parallel to perpendicular to the gradient.  The proof demonstrates that small values of covariance fail to give significance for blocks, and that zero values always give F ratios of 1; therefore, significant variation is entirely derived from covariance. The newly introduced parameter of r indicates the fraction of block to block variation attributable to covariance; let p(F) give the associated probability for r.

There are three possible ways that treatments can vary in combination across replications (aside from the case in which none of them do).  Let us take the case in which only one treatment varies as shown in the graph (see List of Figures with the three hypothetical graphs for F=1).  However this is done, replication differences remain the same as in the error term, F is one, and there is neither significance for replications nor covariance between any treatment pairs. In fact, because the only increase in replication error (BMS) over error mean square derives from covariance, this is the only possible result. Note that for the purposes of this proof, “covariance” is defined as the sum of the product of deviations from each across-block treatment mean (sum of squares) divided by the number of replications less one (number of replication degrees of freedom). The average covariance is the total of such combinations as derived then divided by the number of treatment pairs.

Finally, take the case of near uniform trends across replications. Replication mean square is the sum of both single treatment variations and positive covariance. Perfect uniformity cannot be demonstrated accurately because a positive covariance and an error mean square (EMS) of zero produce significant F ratios approaching infinity. Opposing trends violates assumptions for randomized block design and would be difficult to demonstrate for more than two treatments. This is the instance of negative correlation. It may produce non-significant F values between zero and one. If the effect is exactly opposite to the first treatment, then the two treatment variances are exactly equal to twice their absolute covariance, but the covariance is now negative in sign for F=0 and a randomized block r of -1 (see equations later).

Section 4. Part One of Two: Mathematical Proof Using k=2 Treatments.

Equation 4.1 Block Mean Square Defined Using Sums of Two Treatments. This is the standard textbook formula for BMS where t is the number of treatments and r is the number of replications (not to be confused with the parameters of correlation coefficient or the r ratio derived later). The only difference is that the sum of r-1 blocks is broken down into respective treatments x and y.

Equation 4.2 Expansion of First Square.  

Equation 4.3 Expansion of Second Square.

Equation 4.4 Group Individual Variance and Covariance.

Equation 4.5 Move r To Numerator As Fraction To Complete Defined Forms.

Equation 4.6 Substitute Variance and Covariance Designations.

Equation 4.7 Block Mean Square in Terms of Sum of Variances for t Treatments and the Sum of Covariances for n=1 to C Combinations of Treatment Pairs.

Section 5. Part Two of Two: Proof Using k+1=3 Treatments.

Equation 5.1 Block Mean Square (BMS) Defined Using Three Treatments x, y and z for r blocks.

Equation 5.2 Expansion of First Square.

Equation 5.3 Expansion of Second Square.

Equation 5.4 Grouping for Treatment Variances and Covariances.

Equation 5.5 Move r(r-1) Portion of Denominator to Inner Parentheses.

Equation 5.6 Move r from Denominator to Numerator Fraction to Yield Expressions in Terms of Cross Block Treatment Variance and Covariance.

Equation 5.7 Therefore, Three Treatments Produce the Same Result, BMS Is Composed of Sum of Treatment Variances Plus Twice Sum of Covariance Pairs. Note that for treatment numbers greater than two, the generalized terminology is abbreviated to x and y to represent ALL treatments and treatment pairs. In the subscript for covariance S, S sub xy also implies ALL combinations of treatment pairs no matter how many.

Equation 5.8 Therefore, Covariance is a Component of Block Mean Square.

Section 6. Derivation of Error Mean Square in Terms of Covariance Among Treatments.

EMS (error mean square) can be broken into the above BMS component and sums of variances for treatments across blocks.   The derivation from conventional form for three treatments follows the same pattern as the sum of squares and covariation derivations for BMS in the preceding proof.

The sum of all combinations for treatment variance pairs is for the number, n=1 to C treatments taken 2 at a time (number of treatment pairs). This is the end result for variances across blocks:

Let us use only two treatments to illustrate, having already proved the BMS component for three treatments as x, y and z. ESS (error sum of squares) refers to the conventional sum of squares formula where EMS = ESS divided by (r-1)(t-1), the number of residual degrees of freedom.

Equation 6.1 Conventional Calculation For Error Sum of Squares.

Equation 6.2 Break Sums Into Treatment Components.

Equation 6.3 Expand  Squares.

Note that by careful arrangement of these terms, sums of squares in terms of x and y treatments and covariance between them is isolated, as follows:

Equation 6.4 Group Into Variance-Covariance Forms, Divide By (r-1)(t-1) to Obtain Error Mean Square.

Equation 6.5 Reduce Expression To Conventional Notation. Note that for treatment numbers greater than two, the generalized terminology is abbreviated to x and y to represent ALL treatments and treatment pairs. In the subscript for covariance S, S sub xy also implies ALL combinations of treatment pairs no matter how many.

Equation 6.6 Substituting Equation 8, Section 5 for BMS...

To show that without covariance, F for replications cannot be greater than one, rephrase F by substituting these terms for BMS and EMS into the formula:

Equation 6.7 Conclusion:  F=1 When Covariance=0.

It is seen that the larger covariance is, the greater F will be.  But how large can F be whenever covariance is zero?  If we allow covariance to approach zero, we find that F approaches a lower limit of one.  This is proved by the formula itself.  Let covariance be zero, then the above equation becomes:

Factoring yields:

Therefore, F is exactly one whenever across-block treatment covariance is zero, whatever across block variance may be.  

Section 7. Derivation of Covariance for Treatment Pairs from Equation 6.7.

Equation 7.1 Extending Equations Per Proofs for BMS and EMS With More Than Two Treatments...

Equation 7.2 Multiplying Numerator and Denominator Expressions

Grouping...

By sectioning into separate terms, then dividing numerator and denominator by t...

Dividing both numerator and denominator by t-1...

Equation 7.3 Substituting From Equation 6.5.

The average covariance is the sum divided by the number of combinations; therefore,

Cancelling 2 and (t-1) in the upper fraction...

Equation 7.4 Now, calculate the average covariance for any randomized block design as...

To derive the coefficient of covariance for randomized blocks, r, we start with the following definition. See Statistical Methods, Snedecor and Cochran, Seventh Edition, (1980), page 477, for the possibility of using a probability table for the r value. Note that p(r) should be a function of sine theta where theta is the average angle of intersection of blocks with a gradient. All block to block variation will be comprised of "random" plus covariation among treatments, if any. Note that the randomized block r is NOT the coefficient of correlation used in linear regression, apologies given for any errors or misunderstandings leading to that conclusion. It is, however, a similar ratio that suits our purposes for describing block to gradient intersection (or a lack of gradient uniformity). The r value for block covariance is the fraction of (total) variation for treatments across blocks accounted for by covariance between (all combinations of) treatment pairs (average covariance divided by variance).

Equation 7.5 Finally, r for Correlations Among All Treatment Pairs Across Blocks.

This is a useful parameter, which concludes and encapsulates the point of the theorem, so we give it this distinctive form to represent a parameter for all randomized blocks:

Or, in a more elegant and finished form,

Section 8. Examples.

Assume that we wish to compare gas mileage (mpg) for ten different engine sizes as treatments over a range of driving conditions defined by ten different blocked replications. The most informative and unbiased blocks could be composed of standardized trials done in five different states representing the most varied terrain and traffic conditions. The data collected could be as in the following table:

The column on the far right is the individual treatment variances to be used in the formula for the collective r value derived in this paper. The following standard ANOVA table shows the mean square and F values for the above data set, and in addition the r value. Note that the two values in the above table of 28 and 28 are not consistent with the upward trends in the other nine treatments. If these are changed to 35 and 37, the r value will change from 0.75 to 0.95, indicating the fraction of variation among blocks accountable by covariation among combinations of treatment pairs. The more conventional r ratio using variance cross products in the denominator is included for comparative purposes (in this case, the sum of cross products of all treatment combinations). The conventional r would normally indicate the dependent relation between two sets of observations, and so has no purpose here.

If the changes in values in the tables from block to block are sufficiently consistent, a maximum r value of 1 is obtainable (with corresponding large values of F), as in the following table of values and the matching ANOVA and r (for randomized blocks) value:

A third example demonstrates the possibility of negative covariance, though this violates the assumptions of the randomized block design. An example is restricted to two treatments because additional inverse correlations would tend to cancel. The conventional r in such a case would also have a value of negative one. The following illustrates how perfectly opposing trends in two treatments can result in a theoretical r (for randomized block covariance) value of negative one:

There may be other uses for the combined r values in randomized block designs. Two of four blocks in a field plot might have originally been set at a 45 degree angle to the other two. Separate r values for the two sets could be used to estimate the magnitude and direction of a moisture or other variable, resulting in the better use of only two of the replications and a more precise layout for future experiments. It might also be possible that once a field gradient can be identified and measured, a regression of the randomized block on the gradient can be used to estimate its affect in other experiments.

Since no experiment is perfect, values of less than one will not necessarily represent a poor orientation of blocks; therefore, the above value of 0.75 more likely represents a 25% error among cell values rather than a 25% deviation in block orientation. Still, values of one do represent a perfect orientation, and, unlike the conventional calculation of the correlation coefficient r for differences between two sets of data, the randomized block r is a valid and natural indicator of the degree to which a gradient affects treatment values in a randomized block design. To illustrate this, we now extend the above example to the block to block variation of the hypothetical extraneous variable associated with or causing it. We will also use the results to calculate a valid regression of gas mileage using the extraneous variable.

It might be tempting to assume that the differences in gas mileage among blocks are due to changes in elevation, because the states used for the experiment differ widely in that respect. This would be a bad example, however, because it would be an inverse relationship poorly suited to regression analysis. Also lacking would be a logical explanation as to why altitude would affect gas mileage. A more likely explanation and one worth testing for our example would be the concentration of oxygen, a value which would change with altitude and have a natural relation to the process of combustion. The following table represents an idealized case with one block value differing slightly to prevent division by zero errors in the ANOVA:

The following ANOVA table helps explain these issues:

Rows here no longer represent treatments, because, at the elevation for each blocked replication, all the oxygen values should be identical. The “treatment” F value for the corresponding ANOVA is then expected to be non significant at a value of one. Blocks, of course, would be very highly significant, being the only source of variation among cells, except for the one of 255. Note that the value of 0.99 (less than one due to the single cell value of 255), and not the conventional correlation coefficient, gives us an accurate reflection of the fraction of block to block treatment variation owed to covariance and the fact that our orientation of blocks is perpendicular to the gradient. The conventional value of r as a correlation between mileage and oxygen concentration is compromised by treatment variation owed to engine size. Nevertheless, a nearly perfect correlation between mileage and the blocked gradient of oxygen concentration is represented by the theorem's r for randomized blocks, and the conventional regression would yet be accurate for any particular engine size if the regression is adjusted to include both oxygen and engine size as variables. Since oxygen concentration nearly perfectly matches block variation, blocking can be considered nearly 100 efficient with the oxygen gradient, and the 0.75 value of r being less than 1.00 represents undetermined errors in the measured values (possibly due to unusual atmospheric conditions during testing) and neither a poor orientation of blocks or poor correlation with oxygen concentration.

Section 9. The Equivalence Principle of Non-Uniformity and Random Error - Finding the Gradient Using a Gradius.

The theorem is not an easy concept to grasp, but it provides perhaps the only complete description of the underlying fundamentals of randomized block designs and a means of establishing the characteristics of a gradient. Note that a primary gradient can be broken down into an infinite number of secondary vectors, each with its own direction and magnitude.  As a result, a set of blocks fixed parallel to each other will always be perpendicular to some linear vector of the primary when rotated around an equidistant central pivot. The gradient intersects all blocks equally if to greater or lesser degrees.  It would also always have an r value of one barring sampling errors. If a pivot is not centered so as to be equidistant during rotation, the r values for members of cell (treatment) pairs in the set will alternate in a sinusoidal fashion between one (complete covariance) and zero (no block differences). The same is true for pairs where one member of the set is fixed and the other rotated. This is the key to establishing the gradient direction and degree of uniformity in the experimental area.  Arbitrary scaling can be used to calculate these relationships until the source of the gradient is known, so preliminary plots can be used to find the effective magnitude of the gradient and its influence. That will now be demonstrated.

An r value of one for a set of self parallel blocks indicates only that a primary gradient exists.  It says nothing about the direction or magnitude of the major trend.  If two preliminary blocks were set at right angles in a gradient, their r value could be anything from one to zero depending on the combined orientation to the primary gradient.  The averaged r value for a circular pattern of blocks should then always be zero for pairs of test blocks as one is shifted from zero through 180 degrees from the other. This can be an easy preliminary method of establishing direction and magnitude for any kind of trend. 

Values of r less than one can be seen to result from inconsistencies in the orientation of individual blocks that cause cell values within one block to change at a different rate than those in another block.  A lack of uniformity in response occurs because one block is intersected by the gradient from a different angle than others and treatment values then become skewed.  Whereas differences in cell values within a block may still be arithmetically uniform from end to end, the degree of those differences will change from block to block, and the value of r will diminish because there is no longer perfect uniformity in the gradient as measured by the r values.  Ultimately, uniformity with respect to blocks depends NOT on the existence of a uniform gradient, but that the blocks themselves must be perpendicularly arranged in a consistent fashion to a vector however irregular it might otherwise be. This has far reaching consequences, because “blocks” may consist of elastic dimensions that adhere to a convoluted contour of any kind, even the lining of an intestine or a diurnal cycle, uniformity being defined by the relation between blocks and any chosen contours that have an independent, linear influence on treatment cells.  Keep in mind, too, that data transformations can be used for analysis of this type, so that quadratic or other more common natural relationships can be identified.

While an r value for blocks parallel to each other will theoretically always be one however placed in a uniform gradient, it will always be zero if there is no gradient.  This is an easy way to establish that a gradient exists using only a pair of blocks, but it provides no other information about a primary gradient.  R values less than one owing to irregular gradient intersection cannot be distinguished from any other lack of uniformity or random error, but any significant r value (use p(F)), indicates that a trend exists. There is, fortunately, a method by which the primary gradient can be measured in terms of both magnitude and direction. That is called the gradient wheel.

The Gradient Wheel (Gradius)

The Equivalence Principle of a Uniform Gradient may also be stated by noting that as far as the r value is concerned for a block of treatment values, there is no practical difference between a lack of gradient uniformity (fully random variation among cells) and a less than consistent perpendicular orientation of blocks to the gradient.  Therefore, it is important to test for the presence and influence of a gradient on treatment effects before committing to a final design.  The best way to spoof the presence and direction of a gradient is by using a “wheel,” where three or more blocks of identical treatments (cells) are arranged in a circular fashion around a common central treatment with equidistant treatment values corresponding to radial transects like spokes in a wheel.  This isn't too difficult to do in a geographical space. It can be done with any kind of temporal or spatial cycle.

The wheel for finding direction (and influence) of a uniform gradient can be determined by using identical treatment cells in all transects that intersect blocks and where the gradient effect is transformed to ensure that it begins with zero at the center.  If there are no differences among treatments in the results, then either there is no gradient or the block is intersecting it at right angles (r value is zero).   Such a block placed at right angles to the first will resolve that question, since the cells would show a progression of effect from one end to the other if an extraneous variable were present and each transect would have slopes that vary depending on the deviation of the second pair from first. If the pair is rotated about the gradient, the r value for any pair of perpendicular blocks will vary between one and negative one at 180 degree intervals as the gradient intersects the second block to a varying extent (provided all transects begin with a zero value from center).

The same thing is true for transect slopes if rescaled. Imagine that the first (reference) transect is always parallel to the gradient and its prospective blocks at right angles to it and another initially is superimposed over it, much like the hands of a clock. As the second is rotated relative to the first, the r value (and slope of the second member if properly scaled) between them will vary from one to zero and then to a negative one for each 90 degree rotation in direct proportion to changes in the slope of the second block. Slopes and covariance r values will all vary from zero to one if the slope maximum is rescaled to one and intercepts to zero). These patterns are an excellent check on the assumption of uniformity (or lack of) in the overall experiment. Now we can find both direction and magnitude of any uniform gradient by measuring separate correlation coefficients and slopes for treatment cells in each separate block (regression of treatment cell effects on gradient values). If these cannot be expressed as a function of gradient values (the gradient is not known), distal to proximal positions of the treatment cells in transects with any arbitrary baseline can always be used as placeholders for the unknown gradient values so that the effect of an extraneous variable can always be measured even if the variable is not precisely known.

If a gradient is present in the wheel area, then cell effects in transects should show significant correlation coefficients and regressions where the slopes of the regressions have a clear maximum wherever aligned with the gradient and zero when perpendicular. The slopes and correlation R coefficients will also vary from zero to a maximum in sequence as blocks progressively intersect the gradient at different angles.  If the maximum and minimum transect slope values are rescaled to one and zero, then the sequence of those values as one moves around the wheel should, like r values, approximate a sine wave. The slope values will come in handy for measuring effects on treatments in the final experiment. The r values will have associated F values and probabilities, and the significance of those values is the best test for the presence and direction of a uniform gradient. Even if the gradient is not perfectly uniform, such an analysis can provide excellent feedback on uniformity, direction and magnitude of a gradient. If true gradient values are known, a regression can be used to establish the precise influence of the gradient prior to committing to a final experiment. A decision can then also be made as to whether that relationship (or a transformation of adequate fit) can be factored into the final experiment as an extraneous variable of interest.

If the gradient is approximately uniform, a chart of r values for transect pairs can be readily constructed and used to visualize gradient magnitude and direction. Instead of using the y axis to represent a gradient effect, use data transformations to show spectral gradation or shading from one end of each block to the other.  The transect most parallel to the primary gradient will show a steep transition, while the others will be progressively less steep around the disk or wheel.  The extent to which the gradient is not uniform will be visually evident from deviations in the slopes or r values. If measurements are taken relative to a zero point in the wheel center corresponding to an xy plane, then slopes will vary between a maximum and the negative of the same value. If the centers are transformed to zero and the scaling and range adjusted to match radian increments, then both r values and slopes will show maximums of one and minimums of minus one.

By now, you should be thoroughly confused, so we will insert a diagram that shows how this technique is used. Only a 90 degree range is used. Where a gradient is not yet known, this is not likely to work, because the reference transect must fall on the primary gradient. In practice, 180 degrees should work well, and 360 degrees may be preferable. Keep in mind that the test blocks in the diagram are the thick colored lines that curve from zero to 90 degrees and contain the treatment cells corresponding to values adjusted to match where the corresponding transects intersect from the common zero origin. The Y values (points on the gradient) are scaled to zero at the origin and incremented sine values corresponding to 15 degree increments. The X axis, that is the line perpendicular to the gradient, is scaled identically so that regressions of transects have maximum slopes of one. As difficult as this concept is to grasp at first, with some study it should become understandable. In the diagram, the black r values each correspond to the set of two treatments, the first being the vertical transect that matches the gradient direction and the second the other transect matching a diagonal. Each transect contains the "treatment" cells for five colored blocks arranged radially (this, of course, is NOT a randomized set, as treatment cells are arranged to reflect gradient effects).

It will most helpful to progress through the steps required to arrive at these values. To do that, we will use the same ANOVA technique as in prior examples except that we always use only two treatment sets with the first being zero degrees to the gradient (vertical above). In the table below, the yellow columns represent the normalized "block" data in a gradient wheel. Each transect per row represents a treatment group across blocks. Each row is compared to the first of zero degrees to the gradient for calculating r value, slope, and the correlation R squared for interest. The table is just like those in the example section except that it combines all seven transects and the ANOVA results for the pairs are listed to the right. A regression is also calculated for each pair of treatment sets (the zero angle set always being the first). In this way, the reader can see how the r values and slopes vary as the second member of a set is rotated about the axis. For each such pair, a standard regression is also calculated, the dependent (treatment) variable being the increments on the reference transect, i.e., the one directly in line with the gradient. The table below illustrates the results and corresponds to the same data in the wheel layout diagram above.

As an example, this quarter circle of transects is sufficient, but at least 180 degrees would be required in the field where the gradient angle is not yet known. In this ideal example, the above figures are easy to extend through 180 degrees and the same values would be mirrored from the first 90 in the negative. The below chart of the r values and matching transect slopes for the ideal case is helpful in visualizing the sinusoidal pattern of both extended through 180 degrees.

Remember that slope values for any transect will vary around the clock, so to speak (refer to the wheel figure), from a maximum (one if scaled) at zero degrees from a gradient to zero at 90 degrees, to a negative maximum (or one), and that covariance r values for any pair of blocks perpendicular to each other will also vary between one and negative one for every 90 degrees that the second member of the pair is rotated through the gradient.

It may seem odd that the two curves do not coincide exactly and that the r value curve seems skewed away from the midpoint. This is because there are in effect two angles represented by each pair of transects extending from the wheel center through the set of blocks where each contains the two "treatments" representing the two angles of intersection. The x axis for r values applies to only the second treatment at an angle to the first (which is always fixed parallel to the gradient and vertical in the wheel diagram) for each pair, but the true angle of incidence to the gradient is actually the angle bisecting the two transects used to calculate each r value.

The result is that the effective angle of "incidence" for an r value lags behind the reported angle for the second pair and, therefore, relative to a true sine wave, the curve is skewed towards the extremes of one and negative one. The fact that the two curves are almost, but not quite, superimposed is a fortunate matter, because it can be exploited to give a measure of uniformity for the wheel as a whole. If a linear regression is performed between these two parameters, the R squared for the relation will be equal to 0.89 (it would be different if the angular increment is more or less than 15 degrees). Thus, R squared for a regression of r values plotted against slopes (scaled to a maximum of one) divided by 0.89 yields the relative degree of uniformity for the entire area circumscribed by the test wheel, and the average r value for one half revolution should be a perfect zero. The maximum slope, of course, gives the gradient direction and scaled magnitude. If the gradient identity and true values are measurable, then the true magnitude can also be determined and, if the average probability for matching F values is significant, then the extraneous variable can be included as an independent variable for the final experimental results.

Using the Wheel Configuration for the Final Experiment

This has the advantage of eliminating gradient affects among blocks (average r value equals zero) where gradient affects are not wanted. It has two potential disadvantages. Blocks must be laid out perpendicular to an arbitrary center, but end to end in a radial fashion, so a lot of wasted space may result from such a configuration. The other disadvantage could be that no significant block differences would result and the affect of a gradient would not be measurable, but then this is the advantage where such results are desired. In this configuration the experiment should be treated as a completely randomized and not block experiment since precision would otherwise be lost. It is a more strategic method of eliminating noise than is the completely randomized design and gradient effects should cancel throughout the design.

The Equivalence Principle and a Probability Density Function For r

Let us take each of two cases, first where there is and then where there is not a uniform gradient intersecting blocks.  The conventional experiment and resulting ANOVA does not address this issue or provide specific measures of gradient effects, so we can only be aware of a gradient to the extent that blocks are significant.  However, an r value of zero corresponds to the null hypothesis probability of 100% (1-r) and a value of one corresponds to a probability of zero (1-r) under the assumption that all variation among blocks is due to gradient influence since no other conditions would allow this.  Note also that an r value of zero corresponds to an F value of one with a corresponding p(F) probability of 100% for a null hypothesis, and that as an r value approaches one, the F value approaches infinity with a corresponding p(F) approaching zero. 

It is likely true that all other points of the hypothetical probability function for r correspond to p(F) and also sine theta, even if the existence of any theta or gradient cannot be verified directly. Whether any gradient is discernible or not, its presence is betrayed by significant block differences and non-zero values of r. In practice, little may be known about a gradient and no way to distinguish between a lack of perpendicular orientation and seemingly random (non uniform) variation from block to block.  Unfortunately, I am not at present able to calculate density functions, so must leave that issue up to those more familiar with them. Until then, any value of the randomized block r that is significantly more than zero but less than one indicates the presence of some degree of distortion in the midst of an otherwise uniform field, and it is fair to say that developing a density function for the randomized block r value would be an interesting endeavor.

The Universal Law of r Values and the Michelson-Morley Experiment

It is an interesting fact that the r values for cell pairs in adjacent transects (and slopes for transects) is a universal law that can be applied to any set of vectors. The most common example is the rotation of a bar magnet in an electrical field. The result is the familiar sine wave seen in an oscilloscope for alternating current. Another similar example is the Michelson-Morley interferometer used in the famous experiment that validated Einstein's theory of relativity, although the experiment in this case demonstrated that there was no gradient, i.e., ether as a medium for the transmission of light waves. The results of the experiment are generally reported in textbooks as the comparison of light speed at two 90 degree angles by using an apparatus already known as the Michelson-Morley Interferometer. This may seem off topic, but it is a fact that r values in the randomized block covariance theorem can be applied to any set of data where the possible effect of a gradient is of interest; therefore, the reader's patience is sought.

For the Michelson-Morley experiment, the results could have been erroneous under certain conditions. One possibility is if the direction of travel through the supposed ether happens to be approximately midway between the two 90 degree angles, since the two speeds would then be the same even in an ether. This objection is easily dismissed because of the constant rotation of the earth. But, if the direction of the ether was perpendicular to the base of the interferometer, then the speed of light waves would be equal in all directions measurable in a 360 degree arc. Beyond that, a fairly large range of oblique intersections as the earth revolves on its axis would likely not give figures outside the limits of experimental error unless a significant pattern emerged in the data. This could be done using a version of the gradient wheel and provide r values and F probabilities that give levels of confidence in the results. Nevertheless, in such a case, the ether gradient would be coincident with gravitation, and effects from the moon and other celestial bodies ought to cause a phase difference during rotation of earth. None was observed, but other possibilities exist depending on the ether's structure.

Certainly, at the least, the interferometer apparatus must have been used to compare 90 degree pairs throughout a 360 degree rotation, but then how does one know for sure? A more thorough method would be to calculate r values for pairs of measurements (at distances from the light center) at various angles throughout a 360 degree radius using the gradient wheel method and see whether r values vary between one and negative one as the second member of a pair of transects is rotated about the first. That must be repeated for one or more different axes of revolution in three, not two, dimensional space. This is not meant to challenge the Michelson-Morley results in any way, but only to point out what would happen using this very ingenuous apparatus if there actually were a difference in the speed of light traveling at different directions through an ether relative to the fixed stars - or not. Admittedly, there appears to be no effect of any gradient on light velocity; however, a conducting medium need not necessarily affect velocity.

Establishing the existence and precise direction of an ether using the same interferometer as a gradient wheel would require at least one set of measurements over a complete revolution to find the direction of the horizontal vector, and then repeated for another 360 degrees perpendicular to the first axis. The two r values closest to one should provide the three dimensional coordinates of any gradient relative to the fixed stars (the angle of intersection with a gradient would change with the earth's rotation about both the sun and its own axis). In effect, a three dimensional version of the gradient wheel would be required, or two perpendicular wheels. Unfortunately, the hypothetical estimate of the earth's motion through the ether may not be what has been assumed. To ensure figures within common limits of experimental error, the two measurements might be require precision that is not attainable in practice for reasons dealing with assumptions about the gradient, and this then becomes the focus of a new argument.

Section 10. Relativity and the Unified Field Theory.

However off topic it might seem apply a statistical theorem to advanced physics, the fact is that the study of gradients may exposes logical errors in Einstein's theories, or at least as they depend on interpretations of the Michelson-Morley experiment. These flaws are much more difficult to fix than to spot. There is plenty of reason to consider an ether, if properly defined, just not the giant, disconnected breeze that physicists have used to deny it. The big breeze notion could not have been taken seriously, and that illogical premise could in part be why Einstein's ideas were seized upon. The ether can be easily detected and measured in all sorts of ways, if only indirectly (emphasis, indirectly), and not by means of velocity. Many of these are facts of common experience that have never been questioned since Einstein so befuddled reality. The Michelson-Morley experiment was flawed because the equipment would never have sensed a deflection in velocity under those circumstances. The concept of the gradient "wheel" would still fail to account for an ether under the flawed assumptions ordinarily used to describe it. No experiment is of value without a radical revision in the concept of the nature of the substance that propagates light. That is the issue that must be addressed, NOT the existence of an ether based on relative velocities.

Suggesting that disputing relativity could open a public of closed mind is a sure way to be dismissed, but then there's nothing to lose. To make it simple, dismiss both theories of relativity entirely except for the two fundamental propositions that cannot be denied and are firmly established. Then we can construct the necessary conditions for the ether. The first is the fundamental proposition of the special theory, that there is no correct frame of motion with respect to frames of reference moving with uniform velocity with respect to each other. The laws of physics are identical in either frame, and there is no possible way to correctly ascertain that one is moving more or less than the other, only that they are moving in relation to each other. Einsteins' use of the Lorentz equations defied that postulate. The second proposition is that time is a variable (coincident with linear dimension) with respect to a gravitational field, and that an accelerating (or decelerating) frame of reference produces the equivalent of a gravitational field. If the proper conclusions are drawn consistently from this starting point, then the correct answers should follow.

The nature of the ether should be apparent. Most of the confusion surrounds the use of the Lorentz transformations which make corrections for geometric discrepancies required if the fundamental constant between two inertial frames is a required agreement for the speed of light. This is a very difficult concept, requiring a universally curved space without any common center and folding in on itself in a way that can be conceptualized only in a mystical fashion. Admittedly, the Lorentz transformations do account (seemingly) for discrepancies in Newtonian predictions between observers, but these only occur when interactions take place between two frames of interest as they must when measurements are made, even for a beam of light, and this is the crux of the issue. Kinetic energy, for example, only becomes measurable when two bodies collide, and under such circumstances they are no longer separate inertial frames. The argument most debated is the traveling twin paradox first mentioned by Einstein in 1911 as a consequence of uniform motion. Someone in that audience asked if the one twin wasn't younger only because of the effects of acceleration and deceleration, to which Einstein (unwisely?) replied no, it was only due to uniform motion. Though this argument has been hotly debated and set aside since, it is readily apparent that this statement contradicts the first and very sensible postulate of relativity regarding the equivalence of uniform motion.

Not being a student of advanced physics, the relation between gravitational fields and time constriction was never of serious interest, but there clearly is one that has not been recognized. The Lorentz transformations have been useful in accounting for minute differences in observations that somehow weren't predicted by Newtononian mechanics, but all of the measurements based on inferences about relative velocities out to be explainable by gravitational effects, and this likely pegs Einstein's failure to reconcile his two theories. The nagging feeling that the illogical overhead of a curved space ought not to be required violates the sensibility that Newton provided, leaving others to believe that Einstein was paradoxically brilliant but not understandable except in the case of another rare genius.

Observations of "relativistic" effects do not, unfortunately, follow from the constancy of light speed between moving frames. Instead, they should require only adjustments following from clock speeds that differ after exposure to gravitational fields of variable density, particularly inasmuch as any measurements must involve the same. The traveling twin paradox is one example where there really is no paradox if properly explained. The man in the audience was correct, any difference in ages is due to acceleration or other gravitational effects. The only paradox is that the Lorentz transformations, assuming a non-existent ether as used by Einstein, might seem to match observations. An example is the measured lifetime of muons entering the atmosphere. The muon should experience a time constriction due to its travel towards earth, and as a result it would, as it does, appear to travel farther than expected to a stationary observer. But this does not confirm the Lorentz transformations or require the bending of space-time to explain. Instead, it requires another definition for time and dimension in gravitational fields.

Like all other observations that appear to confirm the theory of relativity using Lorentz transformations in space, the constriction of time in gravitational fields that accounts for them should not violate the fundamental postulates that precede them. Nor should they require that the most fundamental notions about a medium for the transmission of light be discarded simply because the Michelson-Morley experiment failed to find it. Any observations that differ from predictions of Newtonian mechanics should be attributable to an ether that varies in density with respect to ambient gravitational fields. The Lorentz equations are not necessary given a way to measure gravitational fields accurately and independently of the commonly observable effects of acceleration.

While searching for similar points of view by others, I found those by Dr. Paul Marmet to be particularly well stated and not too dissimilar from my own. Here was someone else grappling with the underlying inconsistencies of Einstein's Theory of Relativity, which appears more to be an awkward attempt to fit certain observations than a logical explanation for them. If scientists of high reputation and remarkable achievement can address such matters, then there is hope, or at least there once was. Several days ago, I wrote Dr. Marmet a letter of appreciation using the link on his page (15 August, 2007). I scanned his web site briefly, thinking he was just another eccentric. His son wrote back, sending me his father's obituary and an account that aptly demonstrates the death of science as well as a crime committed by those charged with the responsibility of guarding the freedom of scientific inquiry. Let me here do my own awkward best at honoring Dr. Paul Marmet and report what happened to this courageous man of science at the hands of a modern world that has slipped backwards into darkness. The brief version of the obituary is:

Paul Marmet, Ph. D. (1932-2005)

The account below describes Dr. Marmet's outstanding contributions to the world of science, in particular a sophisticated and useful electron spectrometer that the university powers destroyed because he dared to re-examine the underlying assumptions of relativity...

Marmet and his mentor, Larkin Kerwin, described their pioneer work on this electron source in Citation Classics (Nov. 23, 1987). More than 100 scientific papers of spectroscopic data and interpretations have been published on this subject. Furthermore, about 200 other papers have been presented in numerous international and national meetings. Between 1978 and 1998, the author also published several other papers related to the fundamental principles in physics. Several of these papers are presented on this web site. In 1997-99, physicists of the establishment showed fierce disagreement with the fact that Marmet’s research implied that the fundamental principles of physics were being questioned. Although the experimental work, which could determine the energy of numerous quantum states was highly appreciated and even honored, the physics establishment required that the author should stop questioning the fundamental principles of physics. The author was first informed by NSERC (Natural Science and Engineering Research Council of Canada) to stop doing that fundamental research despite the fact that, being theoretical, it required no research funds - all research grants were used for the experimental work needed for the electron impact apparatus. Since the fundamental research was still going on the following year, the grant was cut to zero, putting an end to experimental work using the monoenergetic electron beams. In May 1999, the head of the physics department came to Marmet’s office and said: “Ce n’est pas ton bureau que nous voulons, ton problème est que tu remets en question les principes fondamentaux de la physique.” (“We do not want your office, your problem is that you keep questioning the fundamental principles of physics.”) Three months later, a letter was sent requiring Marmet's office to become unoccupied before the end of the month. Without research grant and being expelled from his office, Dr. Marmet continued his research alone at home.

This was the irrevocable death of a unique instrument in the world, which was able to measure the electronic structure of negative ions and their ionization efficiency curve using a high resolution monoenergetic electron beam. A few months later, the instrument was destroyed. Also, this shows that physics is not only a science, it is a doctrine. Therefore, there are heretics. It's not different from Galileo’s time!

Since this excursion has brought us so far from the topic of randomized blocks, it will not be further pursued.

Section 11. Conclusion.

Proved: For any set of randomized blocks in a uniform gradient, there exists a number r such that r varies from zero to one as the gradient varies from parallel to perpendicular. All significant differences among replications in a randomized block experiment are attributable to the sum of covariance values for all combinations of treatment pairs across blocks. A combined total and average covariance can be calculated using the F ratio for replications. An averaged r value for randomized blocks for all treatment pairs can also be calculated, so that the significance of replications may be reflected in terms of the fraction of block to block treatment variation owed to covariances across blocks. Ideally, assuming a variable gradient or vector sum of several across blocks is uniform, the average value for treatment pairs would indicate the deviation from right angles that an external gradient intersects replications. If the extraneous variable can be identified and measured, the magnitude and direction of the gradient can also be estimated and used with other regression variables in the experiment for predicting outcomes with better precision. Covariance r values may in certain cases have more practical value than treatment variables in the identication of cause and effect relationships; for example, the etiology of diseases and the identification of pre-existing causal links between disease syndromes.

Literature Cited.

Mendenhall, Introduction to Probability and Statistics, Seventh Edition, (1987), p654, p712

Snedecor and Cochran, Statistical Methods, Seventh Edition, (1980), p255, p256, p264, p477

Blocks (non randomized) Arranged for Perpendicular Intersection...