Is the Null Hypothesis in a Literature Review

Cypher Hypothesis

The zilch hypothesis is a typical statistical theory which suggests that no statistical relationship and significance exists in a set of given single observed variable, between two sets of observed data and measured phenomena.

From: Mineral Exploration , 2013

Statistical and Geostatistical Applications in Geology

Swapan Kumar Haldar , in Mineral Exploration (2nd Edition), 2018

9.2.14 The Null Hypothesis

The zip hypothesis is a characteristic arithmetic theory suggesting that no statistical relationship and significance exists in a ready of given, unmarried, observed variables betwixt two sets of observed data and measured phenomena. The hypotheses play an important function in testing the significance of differences in experiments and between observations. H₀ symbolizes the zippo hypothesis of no difference. It presumes to be true until evidence indicates otherwise. Let u.s. have two sets of manufactory feed argent samples from Table 9.5 and compare the mean grade between fix and population and between ii sets. The zilch hypothesis presumes and states that:

Tabular array 9.5. Average Monthly Manufacturing plant Feed Silvery Grade (thou/t) of a Zinc-Pb-Silver Mine

Sample	Ready I	Prepare II	Departure   (d) Set up (I–II)	(Difference)2
April 2010	49	51	−two	four
May	38	31	7	49
June	33	33	0	0
July	41	41	0	0
August	43	twoscore	3	9
September	52	43	ix	81
October	45	51	−6	36
November	41	32	nine	81
December	36	35	i	1
Jan 2011	33	30	3	nine
N	10	10	ten	10
SUM (Ʃ)	411	387	24	270
$\text{AVG}\left(\overline{\mathrm{10}}\right)$	41.1	38.vii	2.four	27.0
VAR (Stwo)	40.77	61.57	23.lx	1078.67
STD (S)	6.38	vii.85	4.86	32.84

$H_0:{\mathrm{\mu }}_1={\mathrm{\mu }}_0$

where H ₀  =   null hypothesis of no difference, μ_one  =   mean of population 1, and μ₀  =   hateful 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 μ_0. The samples are drawn from the same population such that the variance and shape of the distributions are besides equal. Alternative statistical applications such as t, F, and chi-foursquare tin but refuse a aught hypothesis or fail to refuse it. The evidence can country that the mean of the population from which the samples are drawn does not equal the specified population mean and is expressed as:

$H_0:{\mathrm{\mu }}_{\mathrm{ane}}\ne {\mathrm{\mu }}_0$

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Spatial Distribution

K.K. Borregaard , ... 1000. Nachman , in Encyclopedia of Environmental (Second Edition), 2008

Equally the cypher hypothesis, it is assumed that individuals in a population are randomly distributed among the n sampling units of a sample. If this is the example, it is expected that the variance should equal the boilerplate so that the 'index of dispersion', s ²/x̄, is approximately equal to 1. If the ratio exceeds 1, information technology indicates that the population has a patchy (or clumped) distribution whereas a value less than unity indicates an even (or regular) distribution. However, since information originate from sampling, they volition always be associated with some variation, so it is likely that some divergence in s ²/x̄ from unity will occur even if the underlying distribution is random. Particularly if the sample size due north is minor, s ^two/x̄ will showroom large variation due to sampling dissonance. A χ^two-exam can be used for testing whether due south ⁱⁱ/10̄ deviates significantly from i since χ²= (n−i)s ⁱⁱ/x̄ with northward−one degrees of freedom. It should be noted that the test is 2-tailed (in dissimilarity to the majority of cases where χ^two-tests are used) since values significantly smaller or larger than n−1 can pb to rejection of the nil hypothesis.

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Frequentist Statistical Inference

Daniel S. Wilks , in Statistical Methods in the Atmospheric Sciences (Quaternary Edition), 2019

Chi-Square Test

The chi-foursquare (χ ⁱⁱ) test is a simple and mutual goodness-of-fit examination. It substantially compares a information histogram with the probability distribution (for discrete variables) or probability density (for continuous variables) office. The χ ² examination actually operates more than 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 bachelor for continuous data they are usually more powerful, presumably at to the lowest degree in part because the rounding of information into bins, which may be severe, discards information. However, the χ ^two test is easy to implement and quite flexible, being for case, 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 grade. Regardless of whether the data are detached or continuous, the test statistic involves the counts of data values falling into each class in relation to the computed theoretical probabilities,

(5.xiv) $\begin{array}{l}{\chi }^2=\sum _{\text{classes}}\frac{{\left(\#\text{Observed}-\#\text{Expected}\right)}^{\mathrm{two}}}{\#\text{Expected}}\\ =\sum _{\text{classes}}\frac{{\left(\#\text{Observed}-\mathrm{due\; north}Pr\left\{\text{data in class}\right\}\right)}^2}{\mathrm{northward}Pr\left\{\text{data in form}\right\}}.\end{array}$

In each course, the number (#) of information values expected to occur, according to the fitted distribution, is simply the probability of occurrence in that class multiplied past the sample size, n. This number of expected occurrences demand not be an integer value. If the fitted distribution is very close to the data distribution, the expected and observed counts volition be very shut for each class, and the squared differences in the numerator of Equation 5.14 will all be very minor, yielding a small χ ². If the fit is non practiced, at to the lowest degree a few of the classes volition exhibit large discrepancies. These will be squared in the numerator of Equation 5.14 and lead to big values of χ ^two. It is not necessary for the classes to be of equal width or equal probability, just classes with pocket-size numbers of expected counts should be avoided. Sometimes a minimum of five expected events per class is imposed.

Under the cipher hypothesis that the data were drawn from the fitted distribution, the sampling distribution for the test statistic is the χ ⁱⁱ distribution with parameter ν  =   (# of classes   −   # of parameters fit   −   i) degrees of freedom. The examination will be one-sided, because the examination statistic is confined to positive values by the squaring process in the numerator of Equation five.fourteen, and small values of the test statistic back up H ₀. Correct-tail quantiles for the χ ⁱⁱ distribution are given in Table B.3.

Instance v.4

Comparing Gaussian and Gamma Distribution Fits Using the χ² Test

Consider the gamma and Gaussian distributions as candidates for representing the 1933–82 Ithaca January precipitation data in Table A.two. The approximate maximum likelihood estimators for the gamma distribution parameters (Equations 4.48 or 4.50a, and Equation 4.49) are α  =   three.76 and β  =   0.52   in. The sample mean and standard divergence (i.e., the Gaussian parameter estimates) for these data are ane.96   in. and 1.12   in., respectively. The two fitted distributions are illustrated in relation to the data in Figure 4.16. Table 5.iii contains the information necessary to acquit the χ ² tests for these 2 distributions. The atmospheric precipitation amounts accept been divided into six classes, or bins, the limits of which are indicated in the first row of the table. The second row indicates the number of years in which the January precipitation total was within each grade. Both distributions take been integrated over these classes to obtain probabilities for atmospheric precipitation in each class. These probabilities were so multiplied past n  =   l to obtain the expected number of counts.

Table five.3. The χ ⁱⁱ Goodness-of-Fit Exam Practical to Gamma and Gaussian Distributions for the 1933–82 Ithaca January Precipitation Data

Class	<   i″	i–one.5″	one.five–2″	ii–2.5″	ii.5–three″	>   3″
Observed #	5	16	10	7	7	v
Gamma:
Probability	0.161	0.215	0.210	0.161	0.108	0.145
Expected #	8.05	10.75	ten.50	8.05	five.40	vii.25
Gaussian:
Probability	0.195	0.146	0.173	0.178	0.132	0.176
Expected #	9.75	seven.xxx	eight.65	viii.90	half-dozen.threescore	8.80

Expected numbers of occurrences in each bin are obtained by multiplying the corresponding probabilities by n  =   fifty.

Applying Equation 5.14 yields χ ²  =   5.05 for the gamma distribution and χ ⁱⁱ  =   14.96 for the Gaussian distribution. As was also evident from the graphical comparison in Figure four.16, these test statistics betoken that the Gaussian distribution fits these precipitation data substantially less well. Under the respective null hypotheses, these two exam statistics are drawn from a χ ² distribution with degrees of freedom ν  =   6   −   2   −   ane   =   three; because Table v.3 contains six classes, and two parameters (α and β, or μ and σ, for the gamma or Gaussian, respectively) were fit for each distribution.

Referring to the ν  =   three row of Tabular array B.3, χ ^two  = five.05 is smaller than the 90th percentile value of 6.251, so the nix hypothesis that the data have been drawn from the fitted gamma distribution would non be rejected even at the ten% level. For the Gaussian fit, χ ²  =   fourteen.96 is betwixt the tabulated values of 11.345 for the 99th percentile and xvi.266 for the 99.9th percentile, so this null hypothesis would exist rejected at the i% level, but not at the 0.ane% level. ⋄

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Cardiovascular Toxicology

P.J. Churl , in Comprehensive Toxicology, 2010

6.x.1 Introduction

Morphologic studies of the heart and blood vessels are a critical function of toxicologic investigations. This is true at all stages of such investigations, from the near basic screening studies to detailed investigations of the machinery of deportment of toxins, xenobiotics, and drugs. It is oft the gross or microscopic ascertainment of a chemical effect on the heart or claret vessels that is the first indication of a chemical's adverse furnishings. Morphologic written report 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 together profoundly 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 label of pathogenetic mechanisms, morphologic studies play an important part in toxicologic investigations.

Morphologic methods may well exist some of the least well understood of those methods applied in toxicology or other disciplines. Often, morphologic methods are viewed as confirmatory, or mere add-ons to biochemical, functional, or other experimental protocols and methods. Besides, morphologic methods may be easily and unknowingly misapplied, without proper controls or optimized conditions. Ofttimes, inappropriate or less than adequate morphologic methods are chosen in toxicologic studies. These bug 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 first of this chapter.

6.10.one.1 Elements of a Proper Morphologic Study

It is too frequently incorrectly causeless that morphologic exam of tissues is somehow different from other scientific methodologies, such as biochemical, biophysical, genetic, or epigenetic techniques. Morphologic data collected during a toxicologic investigation is only like whatsoever other form of data and should be treated accordingly. The elements of audio scientific investigation hold for morphologic studies, and these basic elements will be restated with specific reference to morphologic studies.

half-dozen.10.1.1.1 Hypothesis

The essential null hypothesis of most morphologic studies is that the morphology of cells and tissues volition be unaltered by an experimental manipulation. Experiments must be planned and specific methods set out earlier beginning to examine whether statistically significant modify has occurred, that is, to test the underlying hypothesis. When a structural alteration in cells and/or tissues is recognized and proven, this change is termed a lesion (or lesions), which potentially can be correlated to toxin exposure through dose–response and/or time-form studies.

6.10.ane.two Controls

The inclusion of advisable controls is essential to make up one's mind whether a lesion(s) is present, and a lesion is acquired by or associated with an experimental manipulation. Too often, investigators approach the experimental situation with the sole purpose of characterizing a lesion, without fairly examining control specimens for comparison. All appropriate controls must exist included, and in adequate numbers, to make reliable, pregnant comparisons to the experimental. Controls must include groups administered vehicle alone, or whatever treatment/experimental condition that is not the dependent variable under written report. For instance, if a cotreatment is given to protect against (or exacerbate) a toxin-induced lesion, an essential control group given that cotreatment alone must exist analyzed. Otherwise, it would not be known whether the cotreatment itself is toxic.

The normal morphologic structure of the centre and diverse blood vessels has been well described, and many standard references are available (Ferrans 1980; Leak 1980; Rhodin 1974). Nevertheless, reliance on even an practiced's experience to remember all criteria for a normal, unaltered control tissue is not adequate, since fine details of normal control morphology are not always easily recollected, and subtle differences volition arise that crave comparing to the control.

6.x.i.3 Appropriate Choice and Equivalent Handling of Control and Experimental Groups

Control specimens must be identical in all respects to the experimental with regard to animal species, handling and animal husbandry, nutrition, time, and method of sacrifice, and especially fixation and other tissue methods (tissue sampling, processing, sectioning, and staining). It is not proper, for instance, to perform special studies on the experimental tissue without including the control for comparison. Similarly, numbers of controls must be comparable to the experimental.

vi.10.one.four Blinded Ascertainment

Observers who are responsible for ascertaining whether a lesion is present should be blinded to the specimen at the fourth dimension of examination. This is all-time done in a double-blinded fashion, that is, where a third party encodes specimens and the code is not broken until all observations have been made and recorded. Criteria for determining or defining the characteristics of a lesion should be clearly agreed upon at the outset of the study, and the conclusions are strengthened if more one observer is used.

6.10.1.5 Grading of Lesions – Semiquantitative and Quantitative Methods

Sometimes, it is only possible to determine whether lesions are present or absent, without a more than qualitative or quantitative assessment possible. This may exist adequate for addressing the hypothesis as long as it is adamant that such lesions do not occur in controls. An accurate means of determining the severity of lesions is preferable, however. A mutual means of accomplishing this is for all observers to concur on a grading organisation for the lesion; this system should exist recorded and adhered to. Frequently, grading systems for lesions can merely be devised after a lesion is found in preliminary studies. Alternatively, once a lesion is determined to be nowadays in experimental animals, observers may repeat either the whole experiment or the observations on available samples (again, in a blinded fashion).

Typical grading systems might be devised on a numerical scale such as 1–4 or ane–five, as long every bit differences in grades and criteria for each grade are axiomatic and are agreed upon by all observers. Such systems may lend themselves to statistical analysis by Student'southward t-test or ANOVA, given acceptable numbers within groups, but simply within reason (means should not exist taken out to multiple decimal points).

Whenever possible, quantitative rather than qualitative methods should be practical to assess lesions. These methods include counting structures/cell or unit surface area of the specimen, betoken-counting techniques, or any of a variety of available computerized morphometric analytic schemes (Chung et al. 2007; Gong et al. 2008; Ivnitski-Steele et al. 2004). Once again, such methods must exist applied as to control and experimental situations in a blinded style in society to depict pregnant conclusions.

6.ten.i.6 Conducting a Morphologic Evaluation

Morphologic evaluations in toxicologic studies many times may consist of stand-lone experiments whose objective is to ascertain specific pathologic lesions associated with a chemical exposure. Oft, however, morphologic evaluations serve as adjuncts to chemical, genetic, or pathophysiologic experiments, or are performed simultaneously with other studies in preliminary testing when a toxic outcome has non yet been defined. In add-on, morphologic evaluation may be done in order to simply confirm that a cardiovascular specimen is appropriate or adequate for other studies, equally should be done in studies of isolated vessels (Conklin et al. 2006); in such cases, the confirmatory morphologic data may not necessarily be illustrated in the last publication of the written report. No affair what the objectives or specific experimental goals are, the bones principles of sound scientific inquiry in the application of morphologic methods remain unchanged.

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Hypothesis Testing

R. Haines-Young , R. Fish , in International Encyclopedia of Human Geography, 2009

In this case the null hypothesis is that at that place is no difference betwixt dairy farmers and other farmers in the extent to which they recognize nitrates equally a potential pollutant arising from their land. For these data, the differences between the observed and expected counts are much smaller. The Pearson Chi-Square statistic (χ ²) is equal to 1.131, and with one degree of freedom, we would exceed this value to be exceeded by run a risk with a probability of almost 0.29 (i.east., 29 times out of 100). Thus, we cannot pass up the null hypothesis. Dairy farmers announced to be no different to other farmers with respect to nitrates.

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Fundamental Principals of Statistical Inference

Darryl I. MacKenzie , ... James East. Hines , in Occupancy Interpretation and Modeling (Second Edition), 2018

3.5.ane Background and Definitions

In hypothesis testing there is a nothing 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 ( ${\text{H}}_0$ ): $\theta ={\theta }_0$

Alternative Hypothesis ( ${\text{H}}_A$ ): $\theta \ne {\theta }_0$

The culling hypothesis may be one-sided (e.g., $\theta >{\theta }_0$ ) or 2-sided, but often takes the two-sided form presented above. The first step afterwards ane defines the null and alternative hypotheses is to define a test statistic that has a known distribution nether the null hypothesis. The side by side step is to define a disquisitional 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'due south 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 fabricated, depending on the conclusion of whether or not to pass up the zip hypothesis.

•

Type i Mistake: The probability that the cipher hypothesis is rejected when it is true (α).

•

Type two Error: The probability that the nothing hypothesis is not rejected when it is faux (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 ( $1-\beta$ ). A good test will take loftier power even for values of θ close to, but distinct from, ${\theta }_0$ .

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The Part of Information Provision on Public GAP Standard Adoption

D. Jourdain , ... 1000. Shivakoti , in Redefining Variety & Dynamics of Natural Resources Management in Asia, Book one, 2017

18.4.2 Starting time Time Adoption

When testing the goose egg hypothesis H ₀  :=   0, the Wald test statistics of two.59 (Prob(> χ ^two)   =   0.107), and the LR statistic of 0.437 (Prob(> χ ²)   =   0.51) do not allow us to decline 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 disability to reject, estimation results for the individual probit models and for the RBPM are presented in Table 18.5. The 2 formulations are giving similar results in terms of signs of the human relationship between adoption and potential explanatory variables. However, the RBPM formulation is showing college 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 furnishings are reinforcing effects on adoption.

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

Variables	Sep. Probits Coefficients (Robust SD)	RBPM Coefficients (Robust SD)
(intercept)	−   2.35 (0.55)***	−   i.53 (0.61)**
Education (years)	0.09 (0.04)**	0.08 (0.03)**
Experience—medium	0.42 (0.36)	0.38 (0.31)
Experience—high	0.39 (0.36)	0.34 (0.31)
Groups	0.38 (0.29)	0.64 (0.26)**
Gov. contacts—frequent	1.36 (0.27)***	1.21 (0.30)***
Know GAP farmers—high	0.22 (0.31)	0.19 (0.26)
GAP other channels—yes	one.59 (0.39)***	1.43 (0.37)***
Labor per ha	one.85 (0.53)***	i.75 (0.48)***
O-subcontract	0.34 (0.21)	0.29 (0.nineteen)
Full buying	0.fourteen (0.25)	0.15 (0.22)
Exp. toll reductions—yes	1.eighteen (0.24)***	1.01 (0.30)***
Rice training—yes	−   0.26 (0.22)	−   i.23 (0.39)***
Log likelihood	−   75.54
Rice training
Labor per ha	0.34 (0.17)*	0.35 (0.17)**
Perception impact—yeah	0.27 (0.xiv)*	0.27 (0.14)*
Groups—yes	0.92 (0.19)***	0.93 (0.19)***
		0.74 (0.43)
Log likelihood	−   140.vi	−   215.9
N	244	244

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

Likelihood ratio test of ρ  =   0; χ ²(ane)   =   0.59; Prob   > χ ⁱⁱ  =   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 Tabular array eighteen.6. These AME give more meaningful information as they can exist interpreted in terms of impact on the probability of adoption, and besides integrate the potentially indirect effects captured in the recursive system of equation. Results presented in Tabular array 18.half dozen were calculated with the hypothesis of no correlations betwixt the two errors. The variables with the strongest positive impact on Q-GAP adoption are related to subcontract labor available (labha), farmers' affiliations in farmer groups (groups), and farmers connections to sources of information (with an equal strength of the furnishings of extension contacts Govcont and other sources of information GAPcha).

Tabular array 18.half dozen. Average Marginal Effects of the Dependent Variables on Q-GAP Adoption

Variable	Direct	Indirect	Total	St. Err.	Sig. a
QGAP equation
Education (years)	0.016		0.02	0.008	0.047
Feel—medium	0.074		0.07	0.069	0.286
Experience—loftier	0.068		0.07	0.066	0.307
Groups—yes	0.067	0.41	0.48	0.091	0.000
Gov. contacts—frequent	0.290		0.29	0.061	0.000
GAP other channels—yes	0.254		0.25	0.050	0.000
Know GAP farmers—high	0.039		0.04	0.060	0.518
Labor per ha	0.327	0.15	0.48	0.121	0.000
O-farm—yes	0.060		0.06	0.041	0.145
Full buying—yes	0.025		0.03	0.046	0.587
Exp. cost reductions—yep	0.245		0.25	0.053	0.000
Perception bear upon—yes			−   0.004	0.005	0.38
Rice training—yes	−   0.046		−   0.05	0.002	0.000
Training equation
Labor per ha	0.111		0.11	0.060	0.063
Groups—yes	0.278		0.28	0.046	0.000
Perception bear upon—yeah	0.089		0.09	0.046	0.050

a

p value of the Wald examination.

Instruction, measured by the number of schooling years of the household head, has a positive statistically significant relationship. This effect extends to standard adoption the findings of literature on agricultural technologies adoption that the longer the farmers' schooling experience, the college 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.

Too, as the registration likewise 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 express (effectually 2% increase in adoption for an boosted schooling yr).

In the aforementioned 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 non discover consequent 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 instruction were related to conversion to organic rice farming in Surin Province in Thailand. All the same, this link is tenuous in our case. Farmers' participations in associations, cooperatives, and groups take positive and highly significant effects on both grooming and Q-GAP adoption. This confirms results institute for adoption of conservation agriculture (Adesina et al., 2000). In our case, the positive consequence for Q-GAP adoption is just significant under the RBPM formulation, probably as a outcome of the endogeneity of the rice-training variable. Common unobservable variables, such equally dynamism and dedication to rice agriculture, is likely to explain both grooming attendance and Q-GAP adoption. As well, AME results are showing that nigh 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 take a positive and statistically significant touch on 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' power to adjust to changes. Similarly, Moser and Barrett (2006) constitute that learning from extension agents influenced the decision to adopt depression-input rice product methods. More than recently and in a reverse relationship, depression charge per unit of adoption of sustainable agronomics in Red china was linked to inadequate agronomical extension efforts (Liu et al., 2011). A dual human relationship may exist at piece of work: (a) more contacts are improving farmers' skills equally extension officers are transmitting cognition, but on the other hand, the farmers that maintain close contact with extension offices are probably more than dedicated to agriculture. Other channels of information (variable GAPChannel) have also some positive and significant touch on on Q-GAP adoption. Other channels in this case included family members, friends, village chief, customs leader, feel 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 customs members determine to follow similar management patterns as each may not want to be left out (social cohesion). For example, the social cohesion factor was constitute to be one of the primal variables of local community adoption of conservation agriculture in Lao people's democratic republic (Lestrelin et al., 2012). All the same, contrary to our expectations, the number of neighbors known to exist adopting Q-GAP did not take a significant relationship with adoption. Several hypotheses tin can be made about this counterintuitive result. Offset, social cohesion might be low in the agricultural zone we chose. Cardinal plains are now cultivated by relatively larger farms and a substantial number of farms are managed past farmers that do not live permanently in the area and/or are passing orders to contracted labor. As a outcome, the farm-to-farm transmission is probable to exist slower than expected. A 2d and more worrying interpretation for the Q-GAP program would be that farmers that did observe before adopters were not really convinced that it would fit their needs and constraints. Under such an assumption, farmer-to-farmer connections might not exist efficient in spreading the program.

Labor availability is often affecting farmers' adoption decisions, peculiarly for smallholders (White et al., 2005; Lee, 2005). Farmers adopting Q-GAP have to dedicate more than time for rice cultivation. Beginning, 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 than 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 upshot on both adoption and training attendance. Contrary to the variable group, the influence of labha is mainly a directly result, as the fourth dimension constraints are more than likely to be important once Q-GAP has been adopted. Contrary to our expectations, off-farm opportunities accept a positive effect on adoption. However, this relation is not significant and cannot really exist commented on.

Non 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 as well anticipating improve market access for their certified products. Among the different perceptions elicited during the interviews, we retained merely the anticipations farmers had nigh the cost-reduction potential of Q-GAP. The relationship between expected price reduction and adoption was both strong (25% probability increment) and significant, meaning farmers who adopted were really convinced that adopting Q-GAP would reduce their expenditures, possibly through a more rational employ of chemical inputs. A nontested hypothesis here is that participation in Q-GAP could be associated by farmers with defended external advice leading to their more than efficient apply of inputs. Although not included in the model, adopters' expectations were probably high in terms of access to new markets, and cost mark-up (because these variables are positively correlated to the variable on cost-reduction perceptions). We found a positive relationship between farmers attention grooming sessions and their perceived negative impact on the environment, but it is difficult to determine on the "direction" of the relationship. Still, farmers' perception of negative impact on the environment did not translate into a significant upshot on adoption.

The coefficient for training is giving unexpected results every bit we were expecting that farmers having attended a defended presentation near Q-GAP would be more likely to adopt. In fact, both private probit and RBPM models are showing a negative relationship betwixt 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 estimation is that farmers are forming some positive expectations from the unlike contacts they had (justifying their training omnipresence) but are actually disappointed in one case they empathize conspicuously the costs and benefits associated with Q-GAP adoption. On the one hand, farmers may exist expecting college "costs" in terms of labor requirements, which may prevent larger landholders from adopting (identified past the variable labha). On the other hand, some farmers may non 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 tin can only propose that the rice produced under Q-GAP volition be more attractive, but cannot guarantee it. In the aforementioned style, farmers may not exist confident in the chapters of the new practices in reducing production costs (for instance, past using less pesticides or unlike types of fertilizers). In other words, farmers may well exist interested by the general concept of Q-GAP but may ultimately make a rational conclusion related to labor issues as we showed earlier. Finally, one should also note that the negative touch is relatively small (−   v% for farmers attending the training).

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Statistical Methods and Fault Handling

Richard E. Thomson , William J. Emery , in Data Analysis Methods in Physical Oceanography (Third Edition), 2014

3.14.1 Significance Levels and Confidence Intervals for Correlation

One useful application of zilch hypothesis testing is the development of significance levels for the correlation coefficient, r. If we have the nix hypothesis as r  = r _o , where r _o is some approximate of the correlation coefficient, we can determine the rejection region in terms of r at a chosen significance level α for different degrees of liberty (N  −   two). A list of such values is given in Appendix Due east. In that table, the correlation coefficient r for the 95% and 99% significance levels (also chosen the five% 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 case, a sample of 20 pairs of (ten, 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. Information technology is interesting to annotation that, considering of the close relationship between r and the regression coefficient b ₁ of these pairs of values, nosotros could take developed the table for r values using a test of the null hypothesis for b _1.

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

(three.113) $Z_r=\frac{\mathrm{i}}{2}[\mathrm{one}\text{due north}(1+r)-1\text{due north}(1-r)]$

which has the standard fault

(3.114) ${\sigma }_z=\frac{1}{{(N-3)}^{1/2}}$

independent of the value of the correlation. The advisable confidence interval is and so

(3.115) $Z_r-Z_{\alpha /2}{\sigma }_z<Z<Z_r+Z_{\alpha /2}{\sigma }_z$

which tin exist transformed dorsum into values of r using Eqn (iii.113).

Before leaving the subject of correlations we want to stress that correlations are merely statistical constructs and, while nosotros accept 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 the states if we do not apply them carefully. Nosotros again emphasize that a high correlation tin can reveal either a close relationship betwixt ii variables or their simultaneous dependence on a third variable. It is also possible that a loftier correlation may exist due to complete coincidence and accept no causal relationship behind information technology. The bones question that needs to be asked is "does it make sense?" A archetype example (Snedecor and Cochran, 1967) is the high negative correlation (−0.98) betwixt the annual birthrate in Great United kingdom of great britain and northern ireland and the annual product of grunter iron in the United States for the years 1875–1920. This high correlation is statistically meaning for the available N  −   2   =   43 degrees of freedom, but the likelihood of a direct relationship between these ii variables is very low.

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Numerical Ecology

Pierre Legendre , Louis Legendre , in Developments in Environmental Modelling, 2012

3 — Tests of statistical significance

In correlation analysis, the cypher hypothesis H ₀ is usually that the correlation coefficient is equal to zero (i.e. independence of the descriptors). Ane tin also exam the hypothesis that ρ has some particular value other than nil. The full general formula for testing correlation coefficients (for H₀: ρ = 0) is:

(4.39) $F=\frac{r_{jk}^{\mathrm{two}}/{\mathrm{five}}_{\mathrm{ane}}}{\left(1-r_{jg}^2\right)/5_2}$

with five _ane = m and v ₂ = due north – m – 1, where m is the number of variables correlated to j. This F-statistic is compared to the critical value $F_{a\left[v_{\mathrm{i}},5_2\right]}$ . In the case of the elementary correlation coefficient, where m = 1 (in that location is a unmarried variable correlated to j), eq. iv.39 becomes eq. 4.12.

In regression analysis, the goose egg hypothesis is that the coefficient of multiple determination (R²) is aught. To exam the coefficient of multiple decision R ⁱⁱ and the multiple correlation coefficient R, the F-statistic is:

(4.40) $F=\frac{R_{\mathrm{one}.2\dots p}^{\mathrm{two}}/v_1}{\left(\mathrm{one}-R_{\mathrm{i}.2\dots p}^2\right)/v_{\mathrm{ii}}}$

with 5 _one = yard and v _ii = northward – m – ane, where m is the number of explanatory variables; m = p – 1 in the notation of eq. 4.40.

Fractional correlation coefficients are tested in the same manner equally coefficients of simple correlation (eq. 4.12 for the F-examination and eq. 4.13 for the t-test, where v = north – 2), except that i 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 liberty for r_{jk .123} (third-order partial correlation coefficient) is v = (n – 2) – 3 = northward – 5.

This is the same as counting v = due north – m – 1, where m is the number of variables in the model besides j. For fractional correlations, eqs. iv.12 and iv.13 become respectively:

(iv.12) $F=5\frac{r_{jk.1\dots p}^2}{1-r_{j\mathrm{one\; thousand}.\mathrm{i}\dots p}^2}$

and

(iv.13) $t=\sqrt{5}\frac{r_{\text{jk}.1\dots p}}{\sqrt{\mathrm{one}-r_{\text{jk}.\mathrm{one}\dots p}^2}}$

The number of covariables will be called q in Subsections 10.3.5 and 11.ane.seven which describe, respectively, the tests of significance in fractional regression and fractional canonical analysis. Semipartial correlation coefficients are tested using the same Fstatistic equally for partial correlations, as shown in Box 4.ane. As usual (Sections one.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 calculating the probability associated with the computed statistic.

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Postanalysis in Adjustment Computations

Bashar Alsadik , in Adjustment Models in 3D Geomatics and Computational Geophysics, 2019

12.ii Goodness of Fit Test

Goodness of fit examination is an important statistical examination that uses chi-squared χ ² distribution in hypothesis testing of adapted observations and postanalysis.

Every hypothesis test requires preparation of a null hypothesis (H ₀) and an alternative hypothesis (H _a ). If one hypothesis is truthful, the other must be false and vice versa. The nada hypothesis H ₀ ways [57,58]:

–

The observations are like.

–

There is no pregnant correlation betwixt the observed variables.

–

The observations are ordinarily distributed.

On the other hand, the alternative hypothesis H _a means:

–

The observations are different.

–

The correlation is significant betwixt the observations.

–

The information are not commonly distributed.

Two master hypothesis tests are applied with χ ²:

–

When goodness of fit is successfully passed, this indicates that the adjustment of observations trouble is well applied, and no errors be. Statistically, we take the null hypothesis and decline the alternative hypothesis:

(12.1) $H_0:{\hat{\sigma }}_o^2={\sigma }_o^2$

where

σ _o ⁱⁱ: the prior value of the variance (before adjustment) and usually causeless 1.

${\hat{\sigma }}_o^2$ : the posterior value of the variance (afterwards adjustment).

–

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

(12.2) $H_a:{\hat{\sigma }}_o^2\ne {\sigma }_o^{\mathrm{ii}}$

The χ ^two value tin be computed as follows:

(12.3) ${\chi }^2=\frac{r{\hat{\sigma }}_o^2}{{\sigma }_o^2}$

where r is the redundancy in observations.

The thought of the test is to statistically compare the distribution (quality) of observations earlier and after aligning and determine whether the sample data are consistent with a hypothesized distribution. To apply the goodness of fit test, nosotros should specify the significance level α. Frequently, researchers cull 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 χ ² computed value to the χ ² tabular value at a specified significance level as shown in Eq. (12.four):

(12.iv) ${{\chi }^{\mathrm{two}}}_{\left(r,1-\frac{\alpha }{2}\right)}<\frac{r{\hat{\sigma }}_o^2}{{\sigma }_o^{\mathrm{ii}}}<{{\chi }^2}_{\left(r,\frac{\alpha }{2}\right)}$

And so the test indicates a passed value, and the cypher hypothesis H ₀ is accepted.

On the other hand, if:

(12.v) $\left.\begin{array}{c}\frac{r{\hat{\sigma }}_o^2}{{\sigma }_o^2}<{{\chi }^2}_{\left(r,1-\frac{\alpha }{\mathrm{two}}\right)}\\ \frac{r{\hat{\sigma }}_o^2}{{\sigma }_o^2}>{{\chi }^{\mathrm{ii}}}_{\left(r,\frac{\alpha }{2}\right)}\end{array}\right\}$

Then the goodness of fit test fails, and the alternative hypothesis H _o is rejected. In statistics, this kind of examination 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 correct tail of the distribution. And so, when $\frac{r{\hat{\sigma }}_o^{\mathrm{two}}}{{\sigma }_o^2}>{{\chi }^2}_{\left(r,\frac{\alpha }{\mathrm{two}}\right)}$ , H _o is rejected and H _a is accustomed.

Fig. 12.2. Goodness of fit rejection and acceptance levels at 95% conviction.

Accordingly, in this section we briefly presented the goodness of fit examination using χ ² distribution to bespeak a generic impression about the adjustment process. However, this test does not specify which observation has failed or caused the examination to fail, and should be adapted past removal, remeasuring, or reweighing. The post-obit sections will discuss the detection and removal of blundered observations.

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