3. what hypothesis states no equality or existence of differences, relationship, or effect? *

9.2.14 The Null Hypothesis

The null hypothesis is a characteristic arithmetic theory suggesting that no statistical relationship and significance exists in a set of given, single, observed variables between two sets of observed data and measured phenomena. The hypotheses play an important role in testing the significance of differences in experiments and between observations. H0 symbolizes the null hypothesis of no difference. It presumes to be true until evidence indicates otherwise. Let us take two sets of mill feed silver samples from Table 9.5 and compare the mean grade between set and population and between two sets. The null hypothesis presumes and states that:

Table 9.5. Average Monthly Mill Feed Silver Grade (g/t) of a Zinc-Lead-Silver Mine

H0:μ1=μ0

where H0 =  null hypothesis of no difference, μ1 =  mean of population 1, and μ0 =  mean of population 2.

The null hypothesis states that the mean μ1 of the parent population from which the samples are drawn is equal to or not different from the mean of the other population μ0. The samples are drawn from the same population such that the variance and shape of the distributions are also equal. Alternative statistical applications such as t, F, and chi-square can only reject a null hypothesis or fail to reject it. The evidence can state that the mean of the population from which the samples are drawn does not equal the specified population mean and is expressed as:

H0:μ1≠μ0

Chi-Square Test

The chi-square (χ2) test is a simple and common goodness-of-fit test. It essentially compares a data histogram with the probability distribution (for discrete variables) or probability density (for continuous variables) function. The χ2 test actually operates more naturally for discrete random variables, since to implement it the range of the data must be divided into discrete classes, or bins. When alternative tests are available for continuous data they are usually more powerful, presumably at least in part because the rounding of data into bins, which may be severe, discards information. However, the χ2 test is easy to implement and quite flexible, being for example, very straightforward to implement for multivariate data.

For continuous random variables, the probability density function is integrated over each of some number of MECE classes to obtain probabilities for data values in each class. Regardless of whether the data are discrete or continuous, the test statistic involves the counts of data values falling into each class in relation to the computed theoretical probabilities,

(5.14)χ2=∑classes#Observed−#Expected2#Expected=∑classes#Observed−nPrdata in class2n Prdata in class.

In each class, the number (#) of data values expected to occur, according to the fitted distribution, is simply the probability of occurrence in that class multiplied by the sample size, n. This number of expected occurrences need not be an integer value. If the fitted distribution is very close to the data distribution, the expected and observed counts will be very close for each class, and the squared differences in the numerator of Equation 5.14 will all be very small, yielding a small χ2. If the fit is not good, at least a few of the classes will exhibit large discrepancies. These will be squared in the numerator of Equation 5.14 and lead to large values of χ2. It is not necessary for the classes to be of equal width or equal probability, but classes with small numbers of expected counts should be avoided. Sometimes a minimum of five expected events per class is imposed.

Under the null hypothesis that the data were drawn from the fitted distribution, the sampling distribution for the test statistic is the χ2 distribution with parameter ν = (# of classes − # of parameters fit − 1) degrees of freedom. The test will be one-sided, because the test statistic is confined to positive values by the squaring process in the numerator of Equation 5.14, and small values of the test statistic support H0. Right-tail quantiles for the χ2 distribution are given in Table B.3.

6.10.1 Introduction

Morphologic studies of the heart and blood vessels are a critical part of toxicologic investigations. This is true at all stages of such investigations, from the most basic screening studies to detailed investigations of the mechanism of actions of toxins, xenobiotics, and drugs. It is frequently the gross or microscopic observation of a chemical effect on the heart or blood vessels that is the first indication of a chemical’s adverse effects. Morphologic study as a screening tool is only one of its functions, however. Detailed morphologic studies are also employed as accurate measurements of cellular injury in dose–response studies, especially when semiquantitative, quantitative, or morphometric methods are combined with morphologic observation. Furthermore, subcellular studies of organelles and fine structural details add greatly to our understanding of the pathogenetic mechanisms of toxin-induced injury of the cardiovascular system. Thus, at all levels of study, from simple screening to in-depth characterization of pathogenetic mechanisms, morphologic studies play an important role in toxicologic investigations.

Morphologic methods may well be some of the least well understood of those methods applied in toxicology or other disciplines. Frequently, morphologic methods are viewed as confirmatory, or mere add-ons to biochemical, functional, or other experimental protocols and methods. Also, morphologic methods may be easily and unknowingly misapplied, without proper controls or optimized conditions. Often, inappropriate or less than adequate morphologic methods are chosen in toxicologic studies. These issues can be easily avoided if the necessary elements of morphologic studies are adhered to. Therefore, a brief review of the important aspects of applying morphologic methods is warranted at the outset of this chapter.

3.5.1 Background and Definitions

In hypothesis testing there is a null hypothesis and an alternative hypothesis. The idea is to develop a test statistic that has a known distribution under the null hypothesis and see if the observed value of the test statistic based on the data is unusual when compared against this known distribution.

Null Hypothesis (H0): θ=θ0

Alternative Hypothesis (HA): θ≠θ 0

The alternative hypothesis may be one-sided (e.g., θ>θ0) or two-sided, but often takes the two-sided form presented above. The first step after one defines the null and alternative hypotheses is to define a test statistic that has a known distribution under the null hypothesis. The next step is to define a critical region of values for the test statistic where the probability of obtaining values in this region is equal to α (the size of the test).

Hypothesis testing leads to a dichotomous decision; one either rejects the null hypothesis or not. The investigator's decision (to reject or not) can be either right or wrong with respect to the truth. Therefore, statisticians provide a nomenclature for the errors that can be made, depending on the decision of whether or not to reject the null hypothesis.

Type 1 Error: The probability that the null hypothesis is rejected when it is true (α).

Type 2 Error: The probability that the null hypothesis is not rejected when it is false (β).

Power of a Test: The probability that the null hypothesis is rejected when it is false is called the power of the test and is (1−β). A good test will have high power even for values of θ close to, but distinct from, θ0.

18.4.2 First Time Adoption

When testing the null hypothesisH0 := 0, the Wald test statistics of 2.59 (Prob(> χ2) = 0.107), and the LR statistic of 0.437 (Prob(> χ2) = 0.51) do not allow us to reject the null hypothesis of no correlation. However, since the Wald test is close to a 10% threshold, and that outright rejection would be more convincing than inability to reject, estimation results for the individual probit models and for the RBPM are presented in Table 18.5. The two formulations are giving similar results in terms of signs of the relationship between adoption and potential explanatory variables. However, the RBPM formulation is showing higher significance of the relationship with the variables groups and training. Variables included in both equations have the same signs in the two equations, meaning the potentially indirect effects are reinforcing effects on adoption.

Table 18.5. Maximum Likelihood Estimates of Separate and Recursive Probit Models

Wald test of ρ = 0; χ2(1) = 2.59; Prob > χ2 = 0.107.

Likelihood ratio test of ρ = 0; χ2(1) = 0.59; Prob > χ2 = 0.441.

Standard errors in parentheses; *p < 0.1, **p < 0.05, ***p < 0.01.

The AME of each explanatory variable on adoption and training are presented in Table 18.6. These AME give more meaningful information as they can be interpreted in terms of impact on the probability of adoption, and also integrate the potentially indirect effects captured in the recursive system of equation. Results presented in Table 18.6 were calculated with the hypothesis of no correlations between the two errors. The variables with the strongest positive impact on Q-GAP adoption are related to farm labor available (labha), farmers' affiliations in farmer groups (groups), and farmers connections to sources of information (with an equal strength of the effects of extension contacts Govcont and other sources of information GAPcha).

Table 18.6. Average Marginal Effects of the Dependent Variables on Q-GAP Adoption

Education, measured by the number of schooling years of the household head, has a positive statistically significant relationship. This result extends to standard adoption the findings of literature on agricultural technologies adoption that the longer the farmers' schooling experience, the higher the tendency to adopt new technologies (Feder et al., 1985; Chouichom and Yamao, 2010; Liu et al., 2011). Farmers who have been through school are probably more equipped to understand the reason behind Q-GAP efforts and can follow the instructions of the program.

Besides, as the registration also requires participant to record their practices, more educated farmers are probably less impressed by this administrative exercise. However, the magnitude of the relationship is relatively limited (around 2% increase in adoption for an additional schooling year).

In the same way, farmers' experience has a positive (but not significant) relationship with Q-GAP adoption. This is in line with Knowler and Bradshaw (2007) who did not find consistent and clear impacts of experience on adoption of conservation agriculture across the studies they reviewed. If we retain the positive correlation, this indicates that experienced farmers evaluate more positively the potential of the Q-GAP program than inexperienced ones. It concurs with Chouichom and Yamao (2010) who showed that longer experience in farming and more years of education were related to conversion to organic rice farming in Surin Province in Thailand. However, this link is tenuous in our case. Farmers' participations in associations, cooperatives, and groups have positive and highly significant effects on both training and Q-GAP adoption. This confirms results found for adoption of conservation agriculture (Adesina et al., 2000). In our case, the positive effect for Q-GAP adoption is only significant under the RBPM formulation, probably as a result of the endogeneity of the rice-training variable. Common unobservable variables, such as dynamism and dedication to rice agriculture, is likely to explain both training attendance and Q-GAP adoption. Besides, AME results are showing that most of the effect of the variable group on Q-GAP adoption is indirect (via the training variable) reinforcing the possibility of a selection bias.

Frequent contacts with government and extension officers have a positive and statistically significant impact on Q-GAP adoption. This is consistent with the technology adoption literature. Feder et al. (1985), summarizing a large spectrum of adoption studies, concluded that education and extension services contacts improves farmers' ability to adjust to changes. Similarly, Moser and Barrett (2006) found that learning from extension agents influenced the decision to adopt low-input rice production methods. More recently and in a reverse relationship, low rate of adoption of sustainable agriculture in China was linked to inadequate agricultural extension efforts (Liu et al., 2011). A dual relationship may be at work: (a) more contacts are improving farmers' skills as extension officers are transmitting knowledge, but on the other hand, the farmers that maintain close contact with extension offices are probably more dedicated to agriculture. Other channels of information (variable GAPChannel) have also some positive and significant impact on Q-GAP adoption. Other channels in this case included family members, friends, village chief, community leader, experience with GAP for vegetable crops, and local soil doctors (ie, trained volunteers providing soil recommendation services to other farmers in the community).

It was expected that farmers having observed many neighbors adopting Q-GAP would be more likely to adopt: it is very common that community members decide to follow similar management patterns as each may not want to be left out (social cohesion). For example, the social cohesion factor was found to be one of the key variables of local community adoption of conservation agriculture in Laos (Lestrelin et al., 2012). However, contrary to our expectations, the number of neighbors known to be adopting Q-GAP did not have a significant relationship with adoption. Several hypotheses can be made about this counterintuitive result. First, social cohesion might be low in the agricultural zone we chose. Central plains are now cultivated by relatively larger farms and a substantial number of farms are managed by farmers that do not live permanently in the area and/or are passing orders to contracted labor. As a result, the farm-to-farm transmission is likely to be slower than expected. A second and more worrying interpretation for the Q-GAP program would be that farmers that did observe earlier adopters were not really convinced that it would fit their needs and constraints. Under such an assumption, farmer-to-farmer connections might not be efficient in spreading the program.

Labor availability is often affecting farmers' adoption decisions, especially for smallholders (White et al., 2005; Lee, 2005). Farmers adopting Q-GAP have to dedicate more time for rice cultivation. First, it requires recording all activities conducted on the farm and encourages some practices that are likely to substitute time-saving but potentially polluting practices with more knowledge and time-intensive practices. For example, using less pesticides requires more pest monitoring of the rice fields. Not surprisingly the variable labha (ie, the amount of family labor available for rice farming per ha) has a positive and highly significant effect on both adoption and training attendance. Contrary to the variable group, the influence of labha is mainly a direct effect, as the time constraints are more likely to be important once Q-GAP has been adopted. Contrary to our expectations, off-farm opportunities have a positive effect on adoption. However, this relation is not significant and cannot really be commented on.

Not all perception variables could be included in the analysis because of collinearity issues: for example, farmers who anticipated some cost-reduction potential before adopting Q-GAP were also anticipating better market access for their certified products. Among the different perceptions elicited during the interviews, we retained only the anticipations farmers had about the cost-reduction potential of Q-GAP. The relationship between expected cost reduction and adoption was both strong (25% probability increase) and significant, meaning farmers who adopted were really convinced that adopting Q-GAP would reduce their expenditures, possibly through a more rational use of chemical inputs. A nontested hypothesis here is that participation in Q-GAP could be associated by farmers with dedicated external advice leading to their more efficient use of inputs. Although not included in the model, adopters' expectations were probably high in terms of access to new markets, and price mark-up (because these variables are positively correlated to the variable on cost-reduction perceptions). We found a positive relationship between farmers attending training sessions and their perceived negative impact on the environment, but it is difficult to decide on the “direction” of the relationship. However, farmers' perception of negative impact on the environment did not translate into a significant effect on adoption.

The coefficient for training is giving unexpected results as we were expecting that farmers having attended a dedicated presentation about Q-GAP would be more likely to adopt. In fact, both individual probit and RBPM models are showing a negative relationship between training attendance and Q-GAP adoption. This should not be immediately interpreted as a sign of poorly conducted training (although this cannot be ruled out). An equivalently possible interpretation is that farmers are forming some positive expectations from the different contacts they had (justifying their training attendance) but are actually disappointed once they understand clearly the costs and benefits associated with Q-GAP adoption. On the one hand, farmers may be expecting higher “costs” in terms of labor requirements, which may prevent larger landholders from adopting (identified by the variable labha). On the other hand, some farmers may not be convinced about the potential benefits presented to them; as the government agencies are not responsible for the marketing of the Q-GAP rice, they can only suggest that the rice produced under Q-GAP will be more attractive, but cannot guarantee it. In the same way, farmers may not be confident in the capacity of the new practices in reducing production costs (for example, by using less pesticides or different types of fertilizers). In other words, farmers may well be interested by the general concept of Q-GAP but may ultimately make a rational decision related to labor issues as we showed earlier. Finally, one should also note that the negative impact is relatively small (− 5% for farmers attending the training).

3.14.1 Significance Levels and Confidence Intervals for Correlation

One useful application of null hypothesis testing is the development of significance levels for the correlation coefficient, r. If we take the null hypothesis as r = ro, where ro is some estimate of the correlation coefficient, we can determine the rejection region in terms of r at a chosen significance level α for different degrees of freedom (N − 2). A list of such values is given in Appendix E. In that table, the correlation coefficient r for the 95% and 99% significance levels (also called the 5% and 1% levels depending on whether or not one is judging a population parameter or testing a hypothesis) are presented as functions of the number of degrees of freedom.

For example, a sample of 20 pairs of (x, y) values with a correlation coefficient less than 0.444 and N − 2 = 18 degrees of freedom would not be significantly different from zero at the 95% confidence level. It is interesting to note that, because of the close relationship between r and the regression coefficient b1 of these pairs of values, we could have developed the table for r values using a test of the null hypothesis for b1.

The procedure for finding confidence intervals for the correlation coefficient r is to first transform it into the standard normal variable Zr as

(3.113)Zr=12[1n (1+r)−1n(1−r)]

which has the standard error

(3.114)σz=1(N−3)1/2

independent of the value of the correlation. The appropriate confidence interval is then

(3.115)Zr−Zα/2σz<Z<Zr+Zα/2σz

which can be transformed back into values of r using Eqn (3.113).

Before leaving the subject of correlations we want to stress that correlations are merely statistical constructs and, while we have some mathematical guidelines as to the statistical reliability of these values, we cannot replace common sense and physical insight with our statistical calculations. It is entirely possible that our statistics will deceive us if we do not apply them carefully. We again emphasize that a high correlation can reveal either a close relationship between two variables or their simultaneous dependence on a third variable. It is also possible that a high correlation may be due to complete coincidence and have no causal relationship behind it. The basic question that needs to be asked is “does it make sense?” A classic example (Snedecor and Cochran, 1967) is the high negative correlation (−0.98) between the annual birthrate in Great Britain and the annual production of pig iron in the United States for the years 1875–1920. This high correlation is statistically significant for the available N − 2 = 43 degrees of freedom, but the likelihood of a direct relationship between these two variables is very low.

3 — Tests of statistical significance

In correlation analysis, the null hypothesis H0 is usually that the correlation coefficient is equal to zero (i.e. independence of the descriptors). One can also test the hypothesis that ρ has some particular value other than zero. The general formula for testing correlation coefficients (for H0: ρ = 0) is:

(4.39)F=rjk2/v1(1-rjk2)/v2

with v1 = m and v2 = n – m – 1, where m is the number of variables correlated to j. This F-statistic is compared to the critical value Fa[v1,v2]. In the case of the simple correlation coefficient, where m = 1 (there is a single variable correlated to j), eq. 4.39 becomes eq. 4.12.

In regression analysis, the null hypothesis is that the coefficient of multiple determination (R2) is zero. To test the coefficient of multiple determination R2 and the multiple correlation coefficient R, the F-statistic is:

(4.40)F= R1.2…p2/v1(1 -R1.2…p2)/v2

with v1 = m and v2 = n – m – 1, where m is the number of explanatory variables; m = p – 1 in the notation of eq. 4.40.

Partial correlation coefficients are tested in the same way as coefficients of simple correlation (eq. 4.12 for the F-test and eq. 4.13 for the t-test, where v = n – 2), except that one additional degree of freedom is lost for each successive order of the coefficient, or each covariable in the model. For example, the number of degrees of freedom for rjk.123 (third-order partial correlation coefficient) is v = (n – 2) – 3 = n – 5.

This is the same as counting v = n – m – 1, where m is the number of variables in the model besides j. For partial correlations, eqs. 4.12 and 4.13 become respectively:

(4.12)F=vrjk.1…p21- rjk.1…p2

and

(4.13)t=vrjk.1…p1-rjk.1…p2

The number of covariables will be called q in Subsections 10.3.5 and 11.1.7 which describe, respectively, the tests of significance in partial regression and partial canonical analysis. Semipartial correlation coefficients are tested using the same Fstatistic as for partial correlations, as shown in Box 4.1. As usual (Sections 1.2 and 4.2), H0 is tested either by comparing the computed statistic (F or t) to a critical value found in a table for a predetermined significance level α, or by computing the probability associated with the computed statistic.

12.2 Goodness of Fit Test

Goodness of fit test is an important statistical test that uses chi-squared χ2 distribution in hypothesis testing of adjusted observations and postanalysis.

Every hypothesis test requires preparation of a null hypothesis (H0) and an alternative hypothesis (Ha). If one hypothesis is true, the other must be false and vice versa. The null hypothesis H0 means [57,58]:

The observations are similar.

There is no significant correlation between the observed variables.

The observations are normally distributed.

On the other hand, the alternative hypothesis Ha means:

The observations are different.

The correlation is significant between the observations.

The data are not normally distributed.

Two main hypothesis tests are applied with χ2:

When goodness of fit is successfully passed, this indicates that the adjustment of observations problem is well applied, and no errors exist. Statistically, we accept the null hypothesis and reject the alternative hypothesis:

(12.1)H0:σˆo2=σo2

where

σo2: the prior value of the variance (before adjustment) and usually assumed 1.

σˆo2: the posterior value of the variance (after adjustment).

When goodness of fit test fails, this indicates an error in the adjustment procedure such as blunder existence or improper weights assigned to observations. Statistically, we reject the null hypothesis and accept the alternative hypothesis as:

(12.2)Ha:σˆo2≠σo2

The χ2 value can be computed as follows:

(12.3)χ2=rσˆo2σo2

where r is the redundancy in observations.

The idea of the test is to statistically compare the distribution (quality) of observations before and after adjustment and determine whether the sample data are consistent with a hypothesized distribution. To apply the goodness of fit test, we should specify the significance level α. Frequently, researchers choose significance levels equal to 0.01 (confidence of 99%) or 0.05 (confidence of 95%).

After computing the value in Eq. (12.3), we compare the χ2 computed value to the χ2 tabular value at a specified significance level as shown in Eq. (12.4):

(12.4)χ2r1−α2<rσˆo2σo2<χ2rα2

Then the test indicates a passed value, and the null hypothesis H0 is accepted.

On the other hand, if:

(12.5)rσˆo2 σo2<χ2r1−α2rσˆo2σo2>χ2rα2

Then the goodness of fit test fails, and the alternative hypothesis Ho is rejected. In statistics, this kind of test is also called the two tails test (Fig. 12.2). It should be mentioned that, in geomatics, the adjustment tests are only involved with the right tail of the distribution. So, when rσˆo2σo2>χ2r α2, Ho is rejected and Ha is accepted.

Accordingly, in this section we briefly presented the goodness of fit test using χ2 distribution to indicate a generic impression about the adjustment process. However, this test does not specify which observation has failed or caused the test to fail, and should be adapted by removal, remeasuring, or reweighing. The following sections will discuss the detection and removal of blundered observations.