|
|
||
This article is a textbook for those who study the "Meta-Analytic Examination of Assumed Properties of Child Sexual Abuse Using College Samples" (1998) by Bruce Rind, Philip Tromovitch and Robert Bauserman. The essential principles of probability theory, correlation analysis, and statistical tests theory are explained. Among which path analysis, variance analysis, regression analysis, contrast analysis, and sermi-partial correlational analysis are expounded. | ||
by Cyril E. Galaburda,
Dniro City, Ukraine, October 2017
More and more science influences our everyday lives. And purely academical calculations by Bruce Rind, Philip Tromovitch & Robert Bauserman (1998)14 may influence our politics, morals, education, and privacy. As the scientists meta-analyzed 59 studies on child sex abuse
experience in college students' childhood, and it is clear that keeping children in families is much more harmful and dangerous than adult-child sex itself!
What does it mean for us? It means that the right to privacy for children19 tones with the right to privacy for pedophiles20, 1. It means that anti-pedophiliac genocide5 must be stopped, and parents must be imprisoned for keeping children as their property, as slaves17. It means that child anti-sex abuse must become a crime, and if a child wants sex, the sex is by no means abuse.
Of course, Bruce Rind et al. are afraid to make such conclusions. Only pedophiles are in great need of the truth. That's why pedophiles must embalm the Meta-Analysis results. For this purpose they have to understand mathematical statistics and know how to reply to the moralists' insinuations.
My text is a fortification for pedophiles. Its basis is probability theory, its walls are statistical tests. They will withstand any moralists' lie about harmfulness of fornication, and one may fire back with correlation analysis.
I hope that truth will annihilate morals.
Bruce Rind investigates adult-child sex survivors
. There have always been either 13-years-old prostitutes like Baron Corvo's boys, or 13-years-old clients of prostitutes like Federico Fellini, or sex fiends' preys like Mike Tyson, or 10-years-old wives like Mrinolini Debi, or fathers' victims like Joyce Meyer, or 12-years-old teacher-fuckers like Vili Fualaau, or 14-years-old rock groupies like Lori Mattix, or poets'/philosophers'/sultans' minions like Abu Nuwas, or boys picked out for sex by savages like Pueblo Shamans, or young members of the working class
shown in Paris baths to Jean Cocteau, or young porn-watchers like Luis Bunuel, or the others not known to us. The number of all the survivors
of adult-child sex (so called population
) is infinite. What do they have in common with each other?
People believe, all the population shows one or another psychopathologic symptom. If we sample, say, N(sex) people from the population and find psychopathologic symptoms in N(symptom) of them these numbers are supposed to satisfy the equation:
N (symptom)⁄N (sex) = 1.
The left part of this equation may be called conditional frequency of a symptom in adult-child sex survivors
or f (symptom∣sex).
It is a fact that there are adult-child sex survivors
for which
f (symptom∣sex) ≠ 1.
Moralists say that either such samples are too young to show any symptoms, or that psychometry isn't perfect, or that the samples had been enjoying the moral support in their families, or something else18, 14. So we cannot use frequencies and must take another quantity,
Lim[N(sex)→∞] f (symptom∣sex) ≡ P(symptom∣sex),
which is called conditional probability of a symptom in adult-child sex survivors
. Now, this quantity can say something about the population.
If we want to know, for instance, whether family environment reduces symptomatic rates (as moralists say) we should estimate probability of psychopathologic symptoms in different types of samples' families by probability multiplication:
P(family & symptom∣sex) = P(family∣sex) × P(symptom∣sex).
If we want to know whether PTSD is probable for our samples, we should sum up probabilities for all possible symptoms of PTSD,
P(PTSD∣sex) = P(anxiety∣sex) + P(flashbacks∣sex) + P(nightmares∣sex) + P(insomnia∣sex) + P(irritability∣sex) + ..,
like anxiety, flashbacks, nightmares, insomnia, irritability and so on.
Also the more people we study the better we know whether it is probable to have a psychopathologic symptom without experience of adult-child sex:
P(symptom∖sex) = P(symptom) − P(symptom & sex).
(Those who haven't had any adult-child sex experience in childhood are called a control group
.)
If, suppose, having a psychopathologic symptom does not depend on having experience of adult-child sex in childhood (the symptom is observed in adult-child sex survivors
as often as in controls) we may say that having the symptom and having adult-child sex in childhood are independent events
for which:
So if we know probabilities we may obtain whatever we want.
But how to obtain probabilities for our population? There is no way to obtain them. The population is infinite. But still probabilities can be and are used.
For that we must value our samples at different variable quantities. These are: independent variables
of adult-child sex ξ, dependent variables
for different psychopathologic symptoms η, and so called third variables
ζ that characterize adult-child sex survivors'
families at the time of the adult-child sex episode. For instance, we may take ξ = 1 for a sample if (s)he has experience of adult-child sex in childhood and ξ = 0 if not. This person may pass some psychological test and have η points of some psychopathologic symptom. If the person have grown without family problems let be ζ = ζ₁, if the person had been poor in childhood ζ = ζ₂, if the person had been neglected by parents ζ = ζ₃, and so on. These data allow us to identify any sample and to obtain statistics for groups of samples.
For a variable like ϰ we use function of probability distribution
P(ϰ < ϰ₁) ≡ Φμ(ϰ₁)
dependent on some parameter μ. To know this parameter for our population is to know everything about it.
What probability distribution functions are possible? According to the Limiting Theorems, if some variable ϰ₁ = ∑nϰ1n and ∀ϰ1n ≪ ϰ₁ this variable has Normal Distribution
:
Φμ₁,μ₂(ϰ₁) = −∞∫ϰ₁dϰ Exp[−∣ϰ − μ₁∣²⁄(2μ₂)]/√2πμ₂.
Another example. According to Karl Pearson (1900), a sum χ²L ≡ ∑(ℓ ≤ L) ϰ²ℓ every ϰℓ of which has Normal Distribution Φ0,1(ϰℓ) is distributed as:
Φ(χ²L) = [0∫(χ²L)dϰϰL/2−1Exp(−ϰ⁄2)]/[√2Γ(L/2)]
where Γ(…) is the Gamma Function.
There are also Exponential Distribution, Even Distribution, Beta Distribution, Student Distribution and many others, but still we don't know how variables ξ, η and ζ are distributed.
What can we know about them? Any variable ϰ tends to have a value
μ(ϰ) = −∞∫+∞[Φμ(ϰ) − Φμ(ϰ − dϰ)]ϰ
called expected value
, and their difference squared tends to have a value
μ(Δ²ϰ) = μ[∣ϰ − μ(ϰ)∣²]
called variance
.
For instance, a normally-distributed variable ϰ₁ is expected to be μ(ϰ₁) = μ₁, and its μ(Δ²ϰ₁) = μ₂. A Pearson-distributed variable χ²L is expected to be μ(χ²L) = L, and its μ[Δ²(χ²L)] = 2L.
It is easy to calculate that:
where C is some constant.
In the end we know that only expected and variance values of independent, dependent and third variables can say something about population.
We want to study the population of adult-child sex survivors
. Neither Bruce Rind, nor Doctor Laura, nor FBI, nor anyone else can know their number. It is impossible to investigate all of them.
That's why we must sample some part of the population, investigate it and generalize from the part's qualities about the rest of the population.
The most popular adult-child sex survivors
sampling methods are clinical sampling and legal sampling. However data obtained from these samples are not generalizable.
If we are talking about clinical sampling (to investigate only those adult-child sex survivors
who resort to psychiatrical or psychological care) it can be used to prove not only that sex causes psychopathologic symptoms, but also that masturbation causes psychopathologic symptoms. Yes, clinical samples were used by a Swiss physician Samuel Auguste Tissot (1728 – 1797) in showing that masturbation harmed. But according to Sir James Paget (1814 – 1899) Tissot's patients had been inclined to masturbation, not harmed by it. At the same time Tissot had been inclined to attribute psychopathologic symptoms in his patients to their experience of masturbation10, like modern psychiatrists/psychologists are biased to attribute their patients'/clients' psychopathologic symptoms to their adult-child sex experiences16. Also many psychopathologic symptoms in masturbation survivors had been just arrogated to them because Tissot thought that masturbation had caused amnesia10. It is quite similar to modern search for repressed memories
about adult-child sex in psychiatrists' patients and psychologists' clients. If we compare anti-masturbation ideology in XIX century and anti-sex ideology nowadays we may see that clinical sampling is not enough to show that masturbation/sex harms.
Are legal samples (those whose sex with adults have been disclosed by police) enough? According to anti-Rind criticism they aren't. Because police treats minor prostitutes like offenders, not victims; because adult-child sex survivors
are afraid to make their experience known; because none would believe them and so on. That's why legal samples studies are said to be unreliable18. For only ⅟₁₀₀₀ part of American adult-child sex cases and ⅟₃₀₀₀ of European ones are known to police3. Also if we investigate a legal sample we cannot say whether his/her psychopathologic symptoms have been caused by sex itself or by secondary victimization
during parental interrogations, criminal & medical investigations, legal procedures and press intrusion. According to Candidate of Medical Science V. D. Badmaeva for 10% of survivors (of adult-child sex) being under examination or at a trial is a deciding, trigger factor which influences more than sex offence itself
2. It is difficult to know how Ms. Badmaeva detected whether a survivor had been victimized, not secondary victimized, but according to another study in 141 cases… (of adult-child sex a psychiatrist) Körner found that in 10% the child had suffered physical or mental harm from the sexual activity — and in 29.2% harm had been inflicted by the the interrogation
3. So we can see that protecting
children from sex causes psychopathologic symptoms at least as often as sex itself. Therefore legal sampling cannot prove that adult-child sex harms.
If we can use neither clinical, nor legal samples how can we sample adult-child sex survivors
for investigation? A Dutch psychologist Fritz Bernard (1920 – 2006) which have funded scientific study of childlove used to send invitations for potential adult-child sex survivors
throughout the country. This method is called community sampling
and it shows that adult-child sex survivors
in community samples tend to be either normal or only slightly impaired on psychological measures
14.
If we sample from the population so as to generalize from the samples' qualities about all the adult-child sex survivors
of some country it is called national sampling
. There had been studies on British, American and Spanish adult-child sex survivors
. All of them were meta-analysed by Bruce Rind et al. (1997) and shown that adult-child sex survivors
have almost as many psychopathologic symptoms as those who haven't got such experience12.
Community and national samples can really show a lot but for some college professor his/her students are much more handy for study. That's why student samples studies is the largest group of studies on nonclinical populations,.. is useful for addressing questions regarding the general population because about 50% of U. S. adults have some college exposure
14.
Anti-Rind critics objects saying that one cannot generalize from college students' mental resilience, from college students' luck not to have serious
experience, from college students' lack of time to develop psychopathologic symptoms and another college students' qualities about the population of adult-child sex survivors
. But Bruce Rind et al. have shown (2001) that psychopathologic symptoms rates for college, pre-college, former college students and non-students are similar. So college students' qualities are quite generalizable15.
Suppose we take N students and mark those of them who had had adult-child sex in childhood with the value ξ = 1. The rest of them are marked as ξ = 0. This way we get N values of the independent variable:
where ∀in ∈ {1, 2}.
Every n-th student passes some psychological test showing intensity of some psychopathologic symptom, and the student can get either η₁, or η₂, or .., or ηj, or .., or ηJ points:
where ∀jn ∈ {1, 2, .., J}.
Also we may test n-th student's family. If the student have grown without family problems let be ζ = ζ₁, if the student had been poor in childhood ζ = ζ₂, if the student had been neglected by parents ζ = ζ₃, .., if the student had been beaten by parents ζ = ζM. This way we get N values of third variable:
where ∀mn ∈ {1, 2, .., M}.
Do 3N values of our variables prove that adult-child sex causes psychopathologic symptoms? It can be ascertained only through their weighted average values.
For any given variable ϰ ∈ {.., ϰℓ, .., ϰL} its weighted average
〈ϰ〉 is defined as
〈ϰ〉 ≡ ∑(ℓ ≤ L)Cℓϰℓ,
and its coefficients Cℓ are called weights
. If we want them to satisfy equations
the weights must be Cℓ = f(ϰ = ϰℓ) for 〈ϰ〉, and Cℓ = f(ϰ = ϰℓ) × N⁄(N − 1) for 〈Δ²ϰ〉.
Suppose we know averages
and samples variances
for our independent, dependent and third variables. Do these values prove that adult-child sex causes psychopathologic symptoms?
No, they do not. Whether η varies for the reason that Δξ ≠ 0, can be ascertained through their Pearson correlation coefficient
r(ξ,η) ≡ 〈ΔξΔη〉⁄(√〈Δ²ξ〉√〈Δ²η〉).
One may see that absolute value of Pearson correlation coefficient never exceeds one. When r(ξ,η) = ±1.00 our variables ξ and η are in functional dependence on each other which can (according to Taylor Series Theory) be presented as linear: η = A₁ξ + A₀. [Because 〈Δξ×A₁Δξ〉 = A₁×〈Δ²ξ〉 and √〈Δ²(A₁×ξ)〉 = ∣A₁∣×√〈Δ²ξ〉.] The linear dependence means that increasing/decreasing ξ makes η increase/decrease too when r(ξ,η) = A₁⁄∣A₁∣ = 1.00. On the other hand, increasing/decreasing ξ makes η decrease/increase if r(ξ,η) = A₁⁄∣A₁∣ = −1.00.
In the case when r(ξ,η) = 0.00 ⇒ 〈ΔξΔη〉 = 0 we may say that while ξ varies (Δξ ≠ 0) η doesn't vary (Δη = 0), so ξ does not influence η, and the latter cannot be called dependent
variable.
Thus we can say that adult-child sex is concerned with psychopathologic symptoms if r(ξ,η) = 1.00, that psychopathologic symptoms are not caused by adult-child sex if r(ξ,η) = 0.00, that adult-child sex seems to prevent psychopathologic symptoms if r(ξ,η) = −1.00.
What Pearson correlation coefficients do student sampling studies show? For student samples
r(ξ,η) ∈ [−0.25, 0.40].
Adult-child sex and psychopathologic symptoms are almost non-associated. Adult-child sex is unlikely to produce psychopathologic symptoms. For Bruce Rind and Philip Tromovitch results of the meta-analyses indicate that, while about two out of 100 control individuals fall in the clinical range, about three out of 100 CSA individuals will
13. Harmless adult-child sex cannot be called CSA
.
This is not the whole story. We may say that our dependent variable η varies both along the line η = A₁ξ + A₀ and sideways the line:
Δη = Δ(A₁ξ + A₀) + [η − (A₁ξ + A₀)].
According to (**)
because for any variable ϰ its variance is 〈Δ²(ϰ −〈ϰ〉)〉 = 〈Δ²Δϰ〉 = 〈Δ²ϰ〉 and variation [η − (A₁ξ + A₀)] does not depend on variation Δξ, so
(*) ⇒ 〈Δξ × [η − (A₁ξ + A₀)]〉 = 〈Δξ〉 × 〈η − (A₁ξ + A₀)〉 = 0.
Therefore we can see that value
r²(ξ,η) = 〈ΔξΔη〉²⁄(〈Δ²ξ〉〈Δ²η〉) = {〈Δ²η〉 − 〈[η − (A₁ξ + A₀)]²〉}⁄〈Δ²η〉
shows what fraction of the variance 〈Δ²η〉 cannot be attributed to deviating η from the line η = A₁ξ + A₀. The value r²(ξ,η) shows what fraction of the variance 〈Δ²η〉 can be attributed to variation of our independent variable ξ. That's why r²(ξ,η) is coefficient of determining η by ξ.
How much psychopathologic symptoms are determined by adult-child sex experience? For students psychopathologic symptoms are nine times less determined by adult-child sex than by their families:
r²(ζ,η) ≈ 9 × r²(ξ,η)
so keeping children in families is nine times more harmful than sex with adults. Parents, not pedophiles, must be killed first!
Bruce Rind, Philip Tromovitch, and Robert Bauserman have collected more than 54 papers on adult-child sex experience in college students. There they have found N₁ values of Pearson sex-symptoms correlation coefficients like
where ∀kn ∈ {1, 2, .., K} and n ∈ {1, 2, .., N₁}.
Among these N₁ values
Our N₁ values of Pearson sex-symptoms correlation coefficient taken from 54 papers must be averaged. But how? According to Frans Gieles, a correlation coefficient r… is not an interval measure: i. e. the distance between r = 0.1 to r = 0.2 is not the same as the distance from r = 0.8 to r = 0.9
6. Before averaging we must convert our rs into some interval measures zs:
and average them like:
or:
zu = ∑(n ≤ N₁) ½Ln[(1 + rkn)⁄(1 − rkn)] × 1⁄N₁ = ½Ln[(1 + 0.09)⁄(1 − 0.09)],
and get average Pearson coefficient for sex-symptoms correlation:
ru = 0.09
which means that adult-child sex in childhood is barely associated with psychopathologic symptoms in adulthood.
Mathematical statistics cannot be reduced to averaging variables like ϰ. One should test whether a samples average ϰu agrees with the value μ(ϰ) expected in the population. We have to know probability distribution function Φμ(ϰ) in the population of adult-child sex survivors
.
Suppose we've got some sample (of N students or N₁ student sampling studies) for which
These values of ϰ are probable to be distributed either by Φμ(ϰ), or by Φμ′(ϰ), or by Φμ″(ϰ), or by another function. We cannot know exact function since our data are incomplete. If we expand or curtail the data, some distribution functions will seem more probable, and another ones will seem less probable for ϰ in the population.
We need only those distribution functions which probability exceeds some, so called p-value. Bruce Rind et al. (1998) often take p ≡ 0.05, and we say: Probability for ϰ to be distributed by one of the functions {Φμ(ϰ), Φμ′(ϰ), Φμ″(ϰ),..} is at the 0.05 level
.
Such saying is called Null Hypothesis
. Null Hypothesis determines an [(1 − p) × 100% confidence
] interval {μ, μ′, μ″,..}, probability of membership in which for μ(ϰ) exceeds p. If we estimate μ(ϰ) at a value ϰu ∉ {μ, μ′, μ″,..} we should state another hypothesis that is alternative
to Null Hypothesis. The samples data (***) may contradict our Null Hypothesis even when Null Hypothesis is correct. In this case we make an Error of First Kind
which probability is
P(ϰu ∉ {μ, μ′, μ″,..} ∣ Null Hypothesis) = p.
How to know whether our data (***) contradict Null Hypothesis? Sometimes it is enough to see whether positive/negative values of ϰ prevail over the values of the opposite sign in (***). Such prevalence really says something about distribution functions.
Another way is using a quantity (called test statistic
) which distribution function is known for us (for instance, it may be Normal Distribution). Depending on (***) the quantity takes probable or improbable values. If the value is improbable Null Hypothesis is dismissed.
Our Null Hypothesis will be:
Psychopathologic symptoms in adulthood are caused by anything except adult-child sex.
If Null Hypothesis is correct different studies must result in discrepant values of Pearson sex-symptoms correlation coefficient:
Null Hypothesis = (r ∈ [−1.00, 1.00]).
But in reality the resulting unbiased effect size estimate, based
on 15,912 participants, was ru = 0.09, with a 95% confidence interval from 0.08 to 0.11
14. If Null Hypothesis was correct such confidence interval would be less probable than Error of First Kind:
P(ru ∈ [0.08, 0.11] ∣ r ∈ [−1.00, 1.00]) = (0.11 − 0.08)⁄(1.00 − (−1.00)) = 0.015 < 0.050 = p.
That's why we must assume Alternative Hypothesis:
Different studies result in homogeneous values of Pearson correlation coefficient.
But whether our results are really homogeneous? To clarify this we must know that if we've got N₁ values of Pearson sex-symptom correlation coefficients
among which
a sum
has variance
μ{Δ²[N₁×∑(k ≤ K)∣f(r = rk) − P(r = rk ∣ Null Hypothesis)∣²⁄P(r = rk ∣ Null Hypothesis)]} = μ[Δ²(χ²K−1)]
when ∑(k ≤ K)1/P(r = rk ∣ Null Hypothesis) ≪ N₁, and K ≪ N₁11, and
P(r = rk ∣ Null Hypothesis) ≡ P(rk ∈ [0.08, 0.11] ∣ r ∈ [−1.00, 1.00]) = 0.015 or 0.000.
So our test statistic here is χ²K−1 that is supposed to equal (****) if Null Hypothesis is correct. Bruce Rind et al (1998) have calculated that
N₁×∑(k ≤ 54)∣f(r = rk) − P(r = rk ∣ Null Hypothesis)∣²⁄P(r = rk ∣ Null Hypothesis) = 78
which is quite improbable for χ²₅₄₋₁ = χ²₅₃. This value is five times less probable than Error of First Kind:
P{χ²₅₃ ∈ [78, ∞)} = 0.014 < 0.050 = p,
and we must keep our Null Hypothesis.
But Bruce Rind et al. (1998) know how to save rs homogeneity. They declared the values
for which
to be improbable (so called outliers
). If we suppose that
∀kn : 〈z〉 ≫ ½Ln[(1 + rkn)⁄(1 − rkn)] × f(r = rkn)
we may expect that 〈z〉 has Normal Distribution. And it is generally known that probability for any normally-distributed value 〈z〉 to be 1.96 × √〈Δ²〈z〉〉 higher or 1.96 × √〈Δ²〈z〉〉 lower than it's expected value zu is less than 0.05:
P(∣Δ〈z〉∣ > 1.96×√〈Δ²〈z〉〉) ≤ 0.05.
That's why Bruce Rind et al. (1998) ignore the values (*****) and have
N₁×∑(k ≤ 51)∣f(r = rk) − P(r = rk ∣ Null Hypothesis)∣²⁄P(r = rk ∣ Null Hypothesis) = 49.19
which is quite probable for χ²₅₃₋₃ = χ²₅₀:
P{χ²₅₀ ∈ [49.19, ∞)} = 0.506 ≫ 0.050 = p.
That's why Bruce Rind et al. (1998) prefer Alternative Hypothesis to Null Hypothesis.
The fact that adult-child sex correlates with psychopathologic symptoms does not mean that symptoms are caused by sex itself. While correlations may sometimes provide valuable clues in uncovering causal relationships among variables, a non-zero estimated correlation between two variables is not, on its own, evidence that changing the value of one variable would result in changes in the values of other variables. For example, the practice of carrying matches (or a lighter) is correlated with incidence of lung cancer, but carrying matches does not cause cancer
4. In this example fluctuations of independent
(carrying matches) and dependent (cancer) variables are caused by fluctuations of third variable (smoking).
In the same way Bruce Rind et al. (1998) suppose that psychopathologic symptoms (dependent variable) and sex life (independent variable
) are both caused by non-sexual abuse in childhood (third variable). Children are sexual beings only in the case when they've got bad parents.
For me it's rubbish. Mohandas Gandhi married when he was 13-years-old not because he'd got problems with parents. But abusive parents may drive a child to psychopathologic symptoms when they punish him/her for sex.
That's why we may word our Null Hypothesis as:
Family, not sex, causes psychopathologic symptoms,
and Alternative Hypothesis as:
It is better to be poor/ignored/railed/threaten/beaten than satisfied.
These hypotheses weren't tested by Bruce Rind et al. (1998) personally, just taken from another papers on statistical control
. That was accomplished with path analysis, variance analysis and regression analysis.
Let's start from path analysis21. It supposes that fluctuations of our independent, dependent and third variables are associated with points of a directed graph, any influence on some fluctuation is associated with a rib of the graph, and any path in the graph is associated with a linear combination of fluctuations which are supposed to influence each other.
In this model our Null Hypothesis will look like:
Δη = A₂₃ × Δζ + A₂₁ × Δξ,
and Alternative Hypothesis will look like:
Δη ≠ A₂₃ × Δζ + A₂₁ × Δξ.
This linear combination may be divided by square root of dependent variable variance:
Numerators between parantheses are square roots of dependent variable variance when all the other variables but one do not fluctuate:
If we multiply (******) by independent variable fluctuation and average it, we will see from (**) that
r(ξ,η) = [√〈Δ²η〉∣ξ = Const / √〈Δ²η〉] × r(ζ,η) + [√〈Δ²η〉∣ζ = Const / √〈Δ²η〉]
when Null Hypothesis is correct. If we know Pearson coefficients of sex-symptom correlation, family-sex correlation and family-symptom correlation (they are 0.09, 0.13 and 0.29 respectively which means that keeping children in families is much more harmful than sex with them); if we know variances of symptom variable under different values of the other variables, path analysis will show whether Null Hypothesis is really correct. This method shown (1988) that incest-symptom correlations for girls are non-significant without analysing family background of incest8.
That's all I must say about path analysis, and now we'll study variance analysis7. Suppose we've got N₀ students that have grown without family problems (ζ = ζ₁), N₀ students that have been poor in childhood (ζ = ζ₂), N₀ students that have been neglected by parents (ζ = ζ₃), .., N₀ students that have been beaten by parents (ζ = ζM). All the students are sorted into M groups, and intensity of some psychopathologic symptom in them are evaluated as
We may average these values like
or like
μ(η) = ∑(m ≤ M) ∑(n ≤ N₀) ηjmn × 1⁄(MN₀).
Any fluctuation of the symptom variable like
Δηjmn = {∑(m ≤ M) [ηjmn − 〈η〉m]} + {∑(m ≤ M) [〈η〉m − μ(η)]}
contains of varying on account of family and varying on account of another causes. (**) and
∑(n ≤ N₀) [ηjmn − 〈η〉m] = [∑(n ≤ N₀)ηjmn] − N₀ × 〈η〉m = [∑(n ≤ N₀)ηjmn] − N₀ × ∑(n ≤ N₀)ηjmn× 1⁄N₀ = 0
mean that total variance
〈Δ²η〉 = 〈Δ²η∣ζ = Const〉 + 〈Δ²〈η〉〉
is sum of within-group variance
and external variance
, that shows whether family environment influences intensity of symptoms in adult-child sex survivors
.
One may see that
μ[Δ²η∣ζ = Const] = μ[η² − 2 × η × 〈η〉 + 〈η〉²] = μ[〈η〉² − 2 × 〈η〉 × μ(η) + μ²(η)] = μ[Δ²〈η〉],
so while
Δη∣ζ = Const / √μ(Δ²η∣ζ = Const) = Δη∣ζ = Const / √μ(Δ²〈η〉)
and
Δ〈η〉⁄√μ(Δ²〈η〉)
have Normal Distribution Φ0,1
and variances quotient
〈Δ²〈η〉〉 / 〈Δ²η∣ζ = Const〉 has the same distribution function as F(M − 1, MN₀ − N₀) ≡ [χ²M−1 / (M − 1)] / [χ²MN₀−N₀ / (MN₀ − N₀)].
The F is distributed in a predictable way, and may be used as a test statistic which corroborates our Null Hypothesis.
That's all I must say about variance analysis, and now we'll study regression analysis. It supposes the Null Hypothesis to mean
〈η〉 = B₁ × ζ + B₀,
where coefficients
are calculated from empirical data. If empirical data contradict the idea of linear subjection between 〈η〉 and ζ than the Alternative Hypothesis:
〈η〉 = A₁ × ξ + A₀,
may be correct. But it didn't happen.
It had been shown through path analysis, variance analysis and regression analysis that a quarter to all of sex-symptom correlations are caused by non-comfortable family environments of adult-child sex survivors
. That's why living in a family harms, and sex does not harm. Chidren must be protected from parents, not from pedophiles.
Investigating whether sex-symptom correlation coefficients depend on third variables like ϰ we may consider them as independent variables, and the correlation coefficients as dependent variables.
Suppose possible values of ϰ are
and we've got L groups of Pearson sex-symptom correlation coefficient values with N₀ values per group and ϰ = ϰℓ in any ℓth group:
These values may be converted into zs:
averaged
How to know whether these values were influenced by third variable? There are contrast analysis and sermi-partial correlational analysis to clarify it.
Contrast analysis supposes that if ϰ influenced sex-symptom correlation coefficients there would be
In the case of L = 2, for instance, it means that either for ϰ = ϰ₁, or for ϰ = ϰ₂ correlation coefficients would be the same
C₁ × 〈z〉₁ + C₂ × 〈z〉₂ = 〈z〉₁ − 〈z〉₂ = 0 ⇒ 〈z〉₁ = 〈z〉₂,
and ϰ does not influence correlation anyhow.
Whether it is really so should be checked through F-test. Because
That's all I must say about contrast analysis, and now we'll study semi-partial correlational analysis. Semi-partial correlation is the correlation between the dependent variable and the residual of the prediction of one independent variable by the other ones
9, while residual is difference between the predicted independent variable and its prediction through another variables. To predict
is to offer some functional dependence between the variables which (according to Taylor Series Theory) may be presented as linear.
In our case let
Whether such prediction is correct must be checked through F-test again. The value of F conforms to semi-partial correlation coefficient
r[〈z〉, 〈ϰ〉 − (D₁ζ + D₀)] = r[〈ϰ〉 − (D₁ζ + D₀), 〈z〉].
How? Coefficient of determination is expressed as the ratio of the explained variance (variance of the model's predictions…) to the total variance (sample variance of the dependent variable…)
4, and any variance obeys rules (**), so
r²[〈ϰ〉 − (D₁ζ + D₀), 〈z〉] = r²[〈ϰ〉, 〈z〉] + r²[ζ, 〈z〉] = r²[ζ, 〈z〉],
since correlation coefficient may depend on variable ϰ, not on its average 〈ϰ〉 which is constant. That's why Bruce Rind et al. (1998) tried to obtain correlations between each moderator and the effect sizes
14, not between the residual and the effect sizes.
One may see that
This way Bruce Rind et al. (1998) have found that such (third or independent, no matter) value as level of contact during adult-child sex does not influence sex-symptom correlations at all. The value which really influences is level of consent. That's why children can consent to sex (either with themselves, or with another children, or with adults), and if adult-child sex is wanted by the child it is harmless.
Here is what Bruce Rind et al. (1998) Meta-Analysis about. One, who understands this article, may estimate the Meta-Analysis at its true worth.
Bruce Rind may be blamed in pedophilia but passionless numbers cannot be pedophiliac. The American Psychological Association may be forced to repudiate Bruce Rind but χ² cannot be forced to increase. It is possible to prohibit Scott Lilienfeld from writing about Bruce Rind but it is not possible to prohibit the Central Limit Theorem.
Not only common sense but also sexology (see Kinsey Reports), ethology (of Bonobos), ethnography (of Trobriands and another Australoids, of Papuans, of Sumatrans, of Muria, of Lepcha, of Northern Africans, of Hopi & Siriono Indians), history (of Ancient Greece, of Medieval Arabs & Persians, of Western Pre-Industrial societies), biographies (Guy Hocquenghem's, Kirk Douglas'es, Hans van Maanen's, and anothers), and finally mathematical statistics show that sex even with adults is harmless for children. It is the truth that shall make you free.
Consequences of Sexual Abuse in Children and Adolescences,in Russian, Zh. Nevrol., Psikhiatr. Im. S.S. Korsakova, 2009, 109, 12, ISSN 1997-7298, P. 35.
Loving Boys,vol. 2, 1990, ISBN 1-55741-003-08.
Coefficient of determination,from Wikipedia, 2017, URL https://en.wikipedia.org/w/index.php?title=Coefficient_of_determination&oldid=755104809.
Genocide definitions,the Wikipedia, 2015, URL https://en.wikipedia.org/w/index.php?title=Genocide_definitions&oldid=685915956.
Statistical Methods in Education and Psychologyin Russian, Moscow, 1976.
Long-term Effects of Incestuous Child Abuse in College Women,Journal of Consulting and Clinical Psychology, 1988, 56, 5 – 8.
Part (Semi Partial) and Partial Regression Coefficients,in: N. Salkind (ed.), Encyclopedia of Measurement and Statistics, 2007.
Tabu-Zone: Wissenschaftliche Erkenntnisse und etische Grundsätze zum Umgang mit kindlicher Sexualität und Pädophilie, in German, 2013.
Biometrika,1932, vol. 24, P. 351 – 381.
National Samples, Sexual Abuse in Childhood, and Adjustment in Adulthood,Archives of Sexual Behavior, 2006.
The Validity and Appropriateness..,Psychological Bulletin, 2001, vol. 127, No. 6.
The Method of Path Coefficients,The Annals of Mathematical Statistics, vol. 5, No. 3, 1934, Pp. 161 – 215.
|