Chi-Square Goodness-of-Fit Test

The statistical-inference procedures discussed in this thread rely on a distribution called the chi-square distribution. A variable has a chi-square distribution if its distribution has the shape of a special type of right-skewed curve, called a chi-square curve. Actually, there are infinitely many chi-square distributions, and we identify the chi-square distribution in question by its number of degrees of freedom, just as we did for t-distributions.

Basic properties of chi-square curves

• The total area under a chi-square-curve equals 1.
• A chi-square-curve starts at 0 on the horizontal axis and extends indefinitely to the right, approaching, but never touching, the horizontal axis.
• A chi-square-curve is right skewed.
• As the number of degrees of freedom becomes larger, chi-square-curves look increasingly like normal curves.

Chi-Square Goodness-of-Fit Test

Our first chi-square procedure is called the chi-square goodness-of-fit test. We can use this procedure to perform a hypothesis test about the distribution of a qualitative (categorical) variable or a discrete quantitative variable that has only finitely many possible values. Next, let we describe the logic behind the chi-square goodness-of-fit test by an example.

The FBI compiles data on crimes and crime rates and publishes the information in Crime in United States. A violent crime is classified by the FBI as murder, forcible rape, robbery, or aggravated assault. Table 13.1 gives a relative-frequency distribution for (reported) violent crimes in 2010. For instance, in 2010, 29.5% of violent crimes were robberies.

A simple random sample of 500 violent-crime reports from last year yielded the frequency distribution shown in Table 13.2. Suppose that we want to sue the data in Table 13.1 and 13.2 decide whether last year’s distribution of violent crimes is changed from the 2010 distribution.

Solution

The idea behind the chi-square goodness-of-fit test is to compare the observed frequencies in the second column of Table 13.2 to the frequencies that would be expected – the expected frequencies – if last year’s violent-crime distribution is the same as the 2010 distribution. If the observed and expected frequencies match fairly well (i.e., each observed frequency is roughly equal to its corresponding expected frequency), we do not reject the null hypothesis; otherwise, we reject the null hypothesis.

To formulate a precise procedure for carrying out the hypothesis test, we need to answer two questions: 1) What frequencies should we expect from a random sample of 500 violent-crime reports from last year if last year’s violent-crime distribution is the same as the 2010 distribution? 2) How do we decide whether the observed and expected frequencies match fairly well?

The first question is easy to answer, which we illustrate with robberies. If last year’s violent-crime distribution is the same as the 2010 distribution, then, according to Table 13.1, 29.5% of last year’s violent crimes would have been robberies. Therefore, in a random sample of 500 violent-crime reports from last year, we would expect about 29.5% of the 500 to be robberies. In other words, we would expect the number of robberies to be 500 * 0.295, or 147.5.

In general, we compute each expected frequency, denoted E, by using the formula, E = np, where n is the sample size and p is the appropriate relative frequency from the second column of Table 13.1. Using this formula, we calculated the expected frequencies for all four types of violent crime. The results are displayed in the second column of Table 13.3.

The second column of Table 13.3 answer the first question. It gives the frequencies that we would expect if last year’s violent-crime distribution is the same as the 2010 distribution. The second question – whether the observed and expected frequencies match fairly well is harder to answer. We need to calculate a number that measures the goodness-of-fit.

In Table 13.4, the second column repeats the observed frequencies from the second column of Table 13.2. The third column of Table 13.4 reports the expected frequencies from the second column of Table 13.3. To measure the goodness of fit of the observed and expected frequencies, we look at the differences, OE, shown in the fourth column of Table 13.4. Summing these differences to obtain a measure of goodness of fit isn’t very useful because the sum is 0. Instead, we square each difference (shown in the fifth column) and then divided by the corresponding expected frequency. Doing so gives the values (OE)^2 / E, called chi-square subtotals, shown in the sixth column. The sum of the chi-square subtotals, 𝛴(OE)^2 / E = 6.529, is the statistic used to measure the goodness of fit of the observed and expected frequencies.

If the null hypothesis is true, the observed and expected frequencies should be roughly equal, resulting in a small value o the test statistic, 𝛴(OE)^2 / E. As we have seen, that test statistic is 6.529. Can this value be reasonably attributed to sampling error, or is it large enough to suggest that the null hypothesis is false? To answer this question, we need to know the distribution of the test statistic 𝛴(OE)^2 / E.

Pharmacokinetics – Distribution Series

As a result of either direct systemic administration or absorption from an extravascular route, drug reaches the systemic circulation, where it very rapidly distributes throughout the entire volume of plasma water and is delivered to tissues around the body. Two aspects of drug distribution need to be considered: how radidly, and to what extent, the drug in the plasma gets taken up by the tissues. A lot of information on the rate of drug disribution can be obtained by observing the pattern of the changes in the plasma concentrations in the early period following drug administration. Information about the extent of drug distribution can be obtained by considering the value of the plasma concentration once distribution is complete. Thus, the plasma concentration constitutes a "window" for obtaining information on the distribution of the bulk of the drug in the body and how it changes over time.

Extent of Drug Distribution

A drug must reach its site of action to produce an effect. Generally, this involves only a very small amount of the overall drug in the body, and access to the site of action is generally a problem only if the site is located in a specialized area or space. The second important aspect of the extent of drug distribution is the relative distribution of a drug between plasma and the rest of the body. This affects the plasma concentration of the drug and is important because: 1) as discussed above, the plasma concentration is the "window" through which we are able to "see" the drug in the body. It is important to know how a measured plasma concentration is related to the total amount of drug in the body; 2) Drug is delivered to the organs of elimination via the blood. If a drug distributes extensively from the plasma to the tissues, the drug in the plasma will constitute only a small fraction of the drug in the body. Little drug will be delivered to the organs of elimination, and this will hamper elimination. Conversely, it a drug is very limited in its ability to distribute beyond the plasma, a greater fraction of the drug in the body will be physically located in the plasma. The organs of elimination will be well supplied with drug, and this will enhance the elimination processes.

Drug distribution to the tissues is driven primarily by the passive diffusion of free, unbound drug along its concentration gradient. Consider the administration of a single intravenous dose of a drug. In the early period after administration, the concentration of drug in the plasma is much higher than that in the tissues, and there is a net movement of drug from the plasma to the tissues; this period is known as the distribution phase. Eventually, a type of equilibrium is established between the tissues and plasma, at which point the ratio of the tissue to plasma concentration remains constant. At this time the distribution phase is complete and the tissue and plasma concentrations rise and fall in parallel; this period is known as the postdistribution phase. It should be noted that after a single dose, true equilibrium between the tissues and the plasma is not achieved in the postdistribution phase because the plasma concentration falls constinuously as drug is eliminated from the body. This breaks the equilibrium between the two and results in the redistribution of drug from the tissues to the plasma. Uptake and efflux transporters in certain tissues may also be involved in the distribution process and may enhance or limit a drug's distribution to specific tissues.

Physiologic Volumes

Three important physiological volumes – plasma water, extracellular fluid, and total body water, are shown in Figure 4.2. In the systemic circulation, drugs distribute throughout the volume of plasma water (about 3 L). Where a drug goes beyond this, including distribution to the cellular elements of the blood, depends on the physicochemical properties of the drug and the permeability characteristics of individual membranes.

The membranes of the capillary epithelial cells are generally very loose in nature and permit the paracellular passage of even polar and/or large drug molecules. Thus, most drugs are able to distribute throughout the volume of extracellular fluid, a volume of about 15 L. However, the capillary membranes of certain tissues, notably delicate tissues such as the central nervous system, the placenta, and the testes, have much more tightly knit membranes, which may limit the access of certain drugs, particularly large and/or polar drugs.

Once in the extracellular fluid, drugs are exposed to the individual cells of tissues. The ability of drugs to penetrate the membrane of these cells is dependent on a drug's physicochemical properties. Polar drugs and large molecular mass drugs will be unable to pass cell membranes by passive diffusion. However, polar drugs may enter cells if they are substrates for specialized uptake transporters. On the other hand, efflux transporters will restrict the distribution of their substrates. Small lipophilic drugs that can easily penetrate cell membranes can potentially distribute throughout the total body water, which is around 40 L.

In summary, drugs are able to pass through most of the capillary membranes in the body and distribute into a volume approximately equal to that of the extracellular fluid (about 15 L). The ability of a drug to distribute beyond this depends primarily on its physicochemical characteristics. Small, lipophilic drug molecules should penetrate biological membranes with ease and distribute throughout the total body water (about 40 L). A drug's distribution to specific tissues may be enhanced by uptake transporters. Conversely, efflux transporters will restrict the tissue distribution of their substrates. Total body water, about 40 L, represents the maximum volume into which a drug can distribute.

Tissue Binding and Plasma Protein Binding

Given that drug distribution is driven primarily by passive diffusion, it would be reasonable to assume that once distribution has occurred, the concentration of drug would be the same throughout its distribution volume. This is rarely the case because of tissue and plasma protein binding. Drugs frequently bind in a reversible manner to sites on proteins and other macromolecules in the plasma and tissues. At this time it is important to appreciate that bound drug cannot participate in the concentration gradient that drives the distribution process. The bound drug can be considered to be secreted away or hidden in tissue or plasma. Binding has a very important influence on a drug's distribution pattern. Consider a drug that binds extensively (90%) to the plasma proteins but does not bind to tissue macromolecules. In the plasma, 90% of the drug is bound and only 10% is free and able to diffuse to the tissues. At equilibrium, the unbound concentrations in the plasma and tissue will be the same, but the total concentration of drug in the plasma will be much higher than that in the tissues.

Plasma protein binding has the effect of limiting distribution and concentrating drug in the plasma. On the other hand, consider a drug that binds extensively to macromolecules in the tissues but does not bind to the plasma proteins. Assume that overall 90% of the drug in the tissue is bound and only 10% is free. As the distribution process occurs, a large fraction of the drug in the tissues will bind and be removed from participation in the diffusion gradient. As a result, more and more drug will distribute to the tissues. When distribution is complete, the unbound concentrations in the plasma and tissues wil be the same, but the total (bound plus free) average tissue concentration will be much larger than the plasma concentration. Tissue binding essentially draws drug from the plasma and concentrates it in the tissues. Drugs often bind to both the plasma proteins and tissue macromolecules. In this case the final distribution pattern will be determined by which is the dominant process.

Assessment of the Extent of Drug Distribution

Once distribution has gone to completion, the ratio of the total tissue concentration to the total plasma concentration remains constant. The actual tissue concentration (and the ratio) will vary from tissue to tisue, depending on the relative effects of tissue and plasma protein binding. It is not possible to measure individual tissue concentrations, and it is convenient to consider an overall average tissue concentration (Ct). The ratio of Ct to Cp will vary from drug to drug.

It is important to find a way to express a drug's distribution characteristics using a number or distribution parameter that can easily be estimated clinically. The ratio discussed above (Ct/Cp) expresses distribution but cannot be measured easily. Instead, we use the ratio of amount of drug in the body vs. plasma concentration at the same time to express a drug's distribution, that is, the apparent volume of distribution (Vd).

It is important to appreciate that the (apparent) volume of distribution is simply a ratio that has units of volume. It is not physiological volume and, despite its name, it is not the volume into which a drug distributes. The fact that drug A has a Vd value of 20 L does not mean that it distributes into a volume of 20 L, which is greater than extracellular fluid and less than the total body water.

The value of a drug's volume of distribution can be used to estimate the fraction of the drug in the body that is physically present in either the plasma or the tissues. The drug in the body (Ab) may be partitioned into drug in the plasma (Ap) and drug outside the plasma or in the tissues (At):

Ab = Ap + At

the fraction of the drug in the plasma,

fraction in plasma = Ap / Ab

After some algebra, we get

fraction in plasma = Vp / Vd

In a standard 70-kg adult male, Vp = 3 L:

fraction in plasma = 3 / Vd

The fraction of the drug in the body located in the tissues:

fraction in tissue = 1 – fraction in plasma = 1 – 3 / Vd

With this formula we can estimate the fraction of drug in plasma and in tissues, respectively.

Drug in the body is located in either the plasma or the tissues. The amount of drug in either of these spaces is the product of the concentration of drug and the volume of the space. And because Ab = Ap + At, we get

Cp * Vd = Cp *Vp + Ct * Vt

where Cp is the plasma concentration of the drug, Vd the volume of distribution, and Vp the volume of plasma water, Ct the average tissue concentration of the drug, the Vt the overall volume of tissues that the drug distributes.

And because the unbound (free) drug concentration equals the total drug concentration multiplying fraction of unbound, while the unbound drug concentrations between plasma and tissues (extraceullar space) must be the same after reaching distribution equilibrium, we get,

Cp*fu = Ct * fut

After some algebra, we have

Vd = Vp + Vt * fu / fut

where fu is the fraction of unbound drug in plasma and fut is the fraction of unbound drug in tissues. This final equation shows that a drug's volume of distribution is dependent on both the volume into which a drug distributes and on tissue and plasma protein binding. It also shows that increased tissue binding (fut gets smaller) or decreased plasma protein binding (fu gets larger) will result in an increase in the volume of distribution. Also, if a drug binds to neither the plasma proteins (fu = 1) nor the tissues (fut = 1), its volume of distribution will be equal to that of the volume into which the drug distributes (physiologic volume).

Summary

• Vd is a ratio that reflects a drug's relative distribution between the plasma and the rest of the body.
• It is dependent on the volume into which a drug distributes and a drug's binding characteristics.
• It is a constant for a drug under normal conditions.
• Conditions that alter body volume may affect its value.
• Altered tissue and/or protein binding may alter its value.
• It provides information about a drug's distribution pattern. Large values indicate extensive distribution of a drug to the tissues.
• It can be used to calculate the amount of drug in the body if a drug's plasma concentration is known.

Plasma Protein Binding

A very large number of therapeutic drugs bind to certain sites on the proteins in plasma to form drug-protein complexes. The binding process occurs very rapidly, it is completely reversible, and equilibrium is quickly established between the bound and unbound forms of a drug. If the unbound or free drug concentration falls due to distribution or drug elimination, bound drug dissociates rapidly to restore equilibrium. Clinically, although the total drug concentration is measured routinely, pharmacological and toxicological activity is thought to reside with the free unbound drug (Cpu). It is only this component of the drug that is thought to be able to diffuse across membranes to the drug's site of action and to interact with the receptor. Binding is usually expressed using the parameter fraction unbound (fu), and the unbound pharmacologically active component can be calculated:

Cpu = Cp * fu

The three primary plasma proteins combining drugs include albumin, 𝛼1-acid glycoprotein (AAG), and the lipoproteins. AAG is present in lower concentration than albumin and binds primarily neutral and basic drugs. It is referred to as an acute-phase reactant protein because its concentration increases in response to a variety of unrelated stressful conditions, such as cancer, inflammation, and acute myocardial infarction. Given that the unbound concentration is the clinical important fraction and that it is the total concentration that is routinely measured, it is important to know how and when the unbound fraction may change for a drug.

The binding of drug and plasma protein could be regarded as a drug and "receptor" interaction (occupation). So the pharmacodynamic Emax model could be used to describe this interacton mathematically. After some algebra modifications, we get

where PT is the serum concentration of plasma binding protein, Kd the equilibrium dissociation constant, and the Cpu the plasma concentration of unbound (free) drug.

At low concentrations, binding increases in direct proportion to an increase in the free drug (fu remains constant as Cpu increases, where Cpu < Kd). As the free drug concentration increases further, some saturation of the proteins occurs, and proportionally less drug can bind (fu will increase as Cpu increases further). Eventually, at high drug concentrations, all the binding sites on the protein are taken and binding cannot increase further.

The Changes of fu

• Affinity

The affinity of the drug for the protein is the main determinant of fu. Affinity is expressed by Kd, which is a reciprocal form of affinity. As affinity increases, Kd gets smaller. Drugs with small Kd values bind extensively, whereas those with large Kd values will not bind extensively.

• Free drug concentration

Because the therapeutic plasma concentrations of most drugs are much less than their Kd values, binding is able to increase in proportion to increases in the total concentration: fu remains constant over therapeutic plasma concentrations. There are, however, a few drugs that have therapeutic plasma concentrations that are around the range of their Kd values. These drugs, which tend to be drugs that have very high therapeutic plasma concentrations, include valproic acid and salicylates, both of which bind to albumin, and disopyramide, which binds to AAG. The binding of these drugs uses a substantial amount of protein, and as a result they display concentration-dependent binding. As the drug concentration increases, some degree of saturation is observed, and the fraction unbound gets larger.

• Plasma binding protein concentration

As predicted by the law of mass action, changes in the protein concentration will produce changes in the degree of binding. In the case of AAG, increases in the concentration are more common. Physiological stress caused by myocardial infarction, cancer, and surgery can lead to four- to fivefold increases in the AAG concentration. Lipoprotein concentrations vary widely in the population. They can decrease as a result of diet and therapy with HMG-CoA reductase inhibitors (statins), and increase due to alcoholism and diabetes mellitus.

• Displacement

The binding of one drug may displace a second drug from its binding site. This displacement occurs because two drugs compete for a limited number of binding sites on the protein. Not surprisingly, displacers tend to be those drugs that achieve high concentrations in the plasma, use up a lot of protein, and display concentration-dependent binding.

• Renal and hepatic disease

The binding of drugs to ablumin is often decreased in patients with severe renal disease. This appears to be the result of both decreased albumin levels and the accumulation of compounds that are normally eliminated, which may alter the affinity of drugs for albumin and/or compete for binding sites. The binding of several acidic drugs, including phenytoin and valproic acid, is reduced in severe renal disease. Plasma protein binding may also be reduced in hepatic disease.

Clinical Consequences of Changes in Plasma Protein Binding

Changes in fu as a result of altered protein concentration or displacement will result in a change in the fraction of the total drug that is unbound. Two issues need to be addressed when considering the clinical consequences of this: the potential changes in the unbound drug concentration at the site of action, and the interpretation and evaluation of the routinely measured total plasma concentrations.

When binding decreases, the pharmacologically active unbound component increases, and in theory, the response or toxicity could increase. However, the clinical consequences of altered plasma protein binding are minimized by two factors: 1) increased elimination and 2) little change in drug concentrations outside the plasma.

In many cases, only the unbound drug is accessible to the organs of elimination. This is known as restrictive elimination because elimination is restricted by protein binding and is limited to the unbound drug. For drugs display restrictive clearance, the increase in the unbound concentration that occurs when binding decreases results in an increase in elimination of the drug. The increase in elimination is usually proportional to the increase in unbound concentration. As a result, the unbound drug concentration in the plasma eventually falls to exactly the same value as that before the change in binding. In other words, the increase in the unbound concentration is canceled out by increased elimination.

The time it takes for the unbound concentration to return to its normal level is determined by the rate of elimination of the drug (the elimination half-life). If the drug is eliminated rapidly, the unbound concentration returns to its original level quickly. If the drug is eliminated slowly, it takes a long time for the unbound concentration to return to its original level. The time it takes to return can be important for drugs that have a narrow therapeutic index.

The plasma comprises a relative small physiological volume (3 L). Even when plasma protein binding is extensive, the fraction of the drug in the body that is located in the plasma is much less than that in the tissues. As a result, when the fraction unbound increases, the extra drug that distributes to the tissue is often very small in comparison to the amount of drug already present. This is particularly the case for drugs that have large volumes of distribution, where the majority of the drug in the body is in the tissues and only a very small fraction resides in the plasma.

Interpreting Cp

In clinical practice, drug therapy may be monitored by ensuring that plasma concentrations lie within the therapeutic range. The therapeutic range of a drug is expressed most conveniently in terms of concentration routinely measured, the total plasma concentration (Cp). But since the unbound concentration is the pharmacologically active component, the therapeutic range should more correctly be expressed in terms of this unbound concentration. Formulas have been developed for some drugs that will convert a measured plasma concentration of a drug to the value that it would be if the protein concentration were normal. We can prove the below formula with algebra modification.

(when plasma drug concentration << Kd)

Inferences for Population Proportions

Confidence Intervals for One Population Proportion

Statisticicans often need to determine the proportion (percentage) of a population that has a specific attribute. Some examples are:

• the percentage of U.S. adults who have health insurance
• the percentage of cars in the United States that are imports
• the percentage of U.S. adults who favor stricter clean air health standards
• the percentage of Canadian women in the labor force

In the first case, the population consists of all U.S. adults and the specified attribute is "has health insurance." For the second case, the population consists of all cars in the United States and the specific attribute is "is an import." The population in the third case is all U.S. adults and the specified attribute is "favors stricter clean air health standards." In the fourth case, the population consists of all Canadian women and the specified attribute is "is in the labor force."

We know that it is often impractical or impossible to take a census of a large population. In practice, therefore, we use data from a sample to make inferences about the population proportion.

A sample proportion, p^, is computed by using the formula

p^ = x / n

where x denotes the number of members in the sample that have the specified attribute and, as usual, n denotes the sample size. For convenience, we sometimes refer to x as the number of successes and to nx as the number of failures.

The Sampling Distribution of the Sample Proportion

To make inferences about a population mean, 𝜇, we must know the sampling distribution of the sample mean, that is, the distribution of the variable x(bar) (see detail for confidence interval for one population mean at thread "Statistic Procedure – Confidence Interval" http://www.tomhsiung.com/wordpress/2017/08/statistic-procedures-confidence-interval/). The same is true for proportions: To make inferences about a population proportion, p, we need to know the sampling distribution of the sample proportion, that is, the distribution of the variable p^. Because a proportin can always be regarded as a mean, we can use our knowledge of the sampling distribution of the sample mean to derive the sampling distribution of the sample proportion. In practice, the sample size usually is large, so we concentrate on that case.

The accuracy of the normal approximation depdends on n and p. If p is close to 0.5, the approximation is quite accurate, even for moderate n. The farther p is from 0.5, the larger n must be for the approximation to be accurate. As a rule of thumb, we use the normal approximation when np and n(1 – p) are both 5 or greater. Alternatively, another commonly used rule of thumb is that np and n(1 – p) are both 10 or greater; still another is that np(1 – p) is 25 or greater.

Below is the one-proportion z-interval procedure, which is also known as the one-sample z-interval procedure for a population proportion and the one-variable proportion interval procedure. Of note, as stated in Assumption 2 of Procedure 12.1, a condition for using that procedure is that "the number of successes, x, and the number of failures, nx, are both 5 or greater." We can restate this condition as "np^ and n(1 – p^) are both 5 or greather," which, for an unknown p, corresponds to the rule of thumb for using the normal approximation given after Key Fact 12.1.

Determining the Required Sample Size

If the margin of error (E) and confidence level are specified in advance, then we must determine the sample size required to meet those specifications. Solving for n in the formula for margin of error, we get

n = p^(1 – p^)(Z𝛼/2 / E)2

This formula cannot be used to obtain the required sample size because the sample proportion, p^, is not known prior to sampling. There are two ways around this problem. To begin, we examine the graph of p^(1 – p^) versus p^ shown in Figure 12.1. The graph reveals that the largest p^(1 – p^) can be is 0.25, which occurs when p^ = 0.5. The farther p^ is from 0.5, the smaller will be the value of p^(1 – p^). Because the largest possible value of p^(1 – p^) is 0.25, the most conservative approach for determining sample size is to use that value in the above equation. The sample size obtained then will generally be larger than necessary and the margin of error less than required. Nonetheless, this approach guarantees that the specifications will at least be met. In the same vein, if we have in mind a likely range for the observed value of p^, then, in light of Figure 12.1, we should take as our educated guess for p^ the value in the range closest to 0.5. In either case, we should be aware that, if the observed value of p^ is closer to 0.5 than is our educated guess, the margin of error will be larger than desired.

Hypothesis Tests for One Population Proportion

Just earlier, we showed how to obtain confidence intervals for a population proportion. Now we show how to perform hypothesis tests for a population proportion. This procedure is actually a special case of the one-mean z-test. For Key Fact 12.1, we deduce that, for large n, the standardized version of p^,

has approximately the standard normal distribution. Consequently, to perform a large sample hypothesis test with null hypothesis H0: p = p0, we can use the variable

at the test statistic and obtain the critical value(s) or P-value from the standard normal table. We call this hypothesis-testing procedure the one-proportion z-test.

Hypothesis Tests for Two Population Proportions

For independent samples of sizes n1 and n2 from the two populations, we have Key Fact 12.2

Now we can develop a hypothesis-testing procedure for comparing two population proportions. Our immediate goal is to identify a variable that we can use as the test statistic. From Key Fact 12.2, we know that, for large, independent samples, the standardized variabvle z has approximately the standard normal distribution. They null hypothesis for a hypothesis test to compare two population proportions is H0: p1 = p2. If the null hypothesis is true, then p1 – p2 = 0, and, consequently, the bariable in

becomes

However, because p is unknown, we cannot use this variable as the test statistic. Consequently, we must estimate p by using sample information. The best estimate of p is obtained by pooling the data to get the proportion of successes in both samples combined; that is, we estimate p by

Where the p^p is called the pooled sample proportion. After replacing the p by p^p we get the final test statistic, which can be used as the test statsitic and has approximately the standard normal distribution for large samples if the null hypothesis is true. Hence we have Procedure 12.3, the two-proportions z-test. Also, it is known as the two-sample z-test for two population proportions and the two-variable proportions test.

It is very fortunate that the confidence intervals for the difference between two population proportions could be computed. As we can use Key Fact 12.2 to derive a confidence-interval procedure for the difference between two population proportions, called the two-proportions z-interval procedure. Note the following: 1) The two-proportions z-interval procedure is also known as the two-sample z-interval procedure for two population proportions and the two-variable proportions interval procedure. 2) Guidelines for interpreting confidence intervals for the difference, p1p2, between two population proportions are similar to those for interpreting confidence intervals for the difference, 𝜇1 – 𝜇2, between two population means, as describe in other relative threads.

Update on Oct 2 2017

Supplemental Data – Confidence Intervals of Odds Ratio (OR) and Relative Risk (RR)

OR

The sampling distribution of the odds ratio is positively skewed. However, it is approximately normally distributed on the natural log scale. After finding the limits on the LN scale, use the EXP function to find the limits on the original scale. The standard deviation of LN(OR) is

SD of LN(OR) = square root of (1/a + 1/b + 1/c + 1/d)

Now we know the distribution of LN(OR) and the standard deviation (mean and variation) of LN(OR), and the z-proportion procedure could be conducted to compute the confidence intervals of LN(OR).

RR

Similar with OR, the sampling distribution of the relative risk is positively skewed but is approximately normally distributed on the natural log scale. Constructing a confidence interval for the relative risk is similar to constructing a CI for the odds ratio except that there is a different formula for the SD.

SD of LN(RR) = square root of [ b/a(a+b) + d/c(c+d) ]

[Clinical Art][Pharmacokinetics] Interpretation of Plasma Drug Concentrations (Steady-State)

Plasma drug concentration are measured in the clinical setting to determine whether a potentially therapeutic or toxic concentration has been produced by a given dosage regimen. This process is based on the assumption that plasma drug concentrations reflect drug concentrations at the receptor and, therefore, can be correlated with pharmacologic response. This assumption is not always valid. When plasma samples are obtained at inappropriate times or when other factors (such as delayed absorption or altered plasma binding) confound the usual pharmacokinetic behavior of a drug, the interpretation of serum drug concentrations can lead to erroneous pharmacokinetic and pharmacodynamic conclusions and utimately inappropriate patient care decisions. These facors are discussed below.

Confounding Factors

To properly interpret a plasma concentration, it is essential to know when a plasma sample was obtained in relation to the last dose administered and when the drug regimen was initiated.

• If a plasma sample is obtained before distribution of the drug into tissue is complete, the plasma concentration will be higher than predicted on the basis of dose and response. (avoidance of distribution)
• Peak plasma levels are helpful in evaluating the dose of antibiotics used to treat severe, life-threatening infections. Although serum concentrations for many drugs peak 1 to 2 hours after an oral dose is administered, factors such as slow or delayed absorption can significantly delay the time at which peak serum concentrations are attained. Large errors in the estimation of Css max can occur if the plasma sample is obtained at the wrong time. Therefore, with few exceptions, plasma samples should be drawn as trough or just before the next dose (Css min) when determining routine drug concentration in plasma. These trough levels are less likely to be influenced by absorption and distribution problems. (slow or delayed absorption)
• When the full therapeutic response of a given drug dosage regimen is to be assessed, plasma samples should not be obtained until steady-state concentrations of the drug have been achieved. If drug doses are increased or decreased on the basis of drug concentrations that have been measured while the drug is still accumulating, disastrous consequences can occur. Nevertheless, in some clinical situations it is appropriate to measure drug levels before steady state has been achieved. If possible, plasma samples should be drawn after a minimum of two half-lives beause clearance values calculated from drug levels obtained less than one half-life after a regimen has been initiated are very sensitive to small differences in the volume of distribution and minor assay errors. (Whether steady-state attained)
• The impact of drug plasma protein binding on the interpretation of plasma drug coencentration has been discussed in thread "The Plasma Protein Concentration and The Interpretation of TDM Report" before.

Revising Pharmacokinetic Parameters

The process of using a patient's plasma drug concentration and dosing history to determine patient-specific pharmacokinetic parameters can be complex and difficult. A single plasma sample obtained at the appropriate time can yield information to revise only one parameter, either the volume of distribution or clearance, but not both. Drug concentrations measured from poorly timed samples may prove to be useless in estimating a patient's V or Cl values. Thus, the goal is to obtain plasma samples at times that are likely to yield data that can be used with confidence to estimate pharmacokinetic parameters. In addition, it is important to evaluate available plasma concentration data to determine whether they can be used to estiamte, with some degree of confidence, V and/or Cl. The goal in pharmacokinetic revisions is not only to recognize which pharmacokinetic parameter can be revised, but also the accuracy or confidence one has in the revised or patient-specific pharmacokinetic parameter. In the clinical setting, based on the way drugs are dosed and the recommended time to sample, bioavailability is almost never revised, volume of distribution is sometimes revised, and most often clearance is the pharmacokientic parameter that can be revised to determine a patient-specific value.

Volume of Distribution

A plasma concentration that has been obtained soon after administration of an initial bolus is primarily determined by the dose administered and the volume of distribution. This assumes that both the absorption and distribution phases have been avoided.

C1 = (S) (F) (Loading Dose) x e(-kt1) / V (IV Bolus Model)

When e(-kt1) approches 1 (i.e., when t1 is much less than t1/2), the plasma concentration (C1) is primarily a function of the administered dose and the apparent volume of distribution. At this point, very little drug has been eliminated from the body. As a clinical guideline, a patient's volume of distribution can usually be estimated if the absorption and distribution phase are avoided and t1, or the interval between the administration and sampling time, is less than or equal to one-third of the drug's half-life. As t1 exceeds one-third of a half-life, the measured concentration is increasingly infuenced by clearance. As more of the drug is eliminated (i.e., t1 increases), it is difficult to estimate the patient's V with any certainty. The specific application of this clinical guideline depends on the confidence with which one knows clearance. If clearance is extremely variable and uncertain, a time interval of less than one-third of a half-life would be necessary in order to revise volume of distribution. On the other hand, if a patient-specific value for clearance has already been determined, then t1 could exceed one-third of a half-life and a reasonably accurate estimate of volume of distribution could be obtained. It is important to recognize that the pharmacokinetic parameter that most influences the drug concentration is not determined by the model chosen to represent the drug level. For example, even if the dose is modeled as a short infusion, the volume of distribution can still be the important parameter controlling the plasma concentration. V is not clearly defined in the equation (see it below); nevertheless, it is incorporated into the elimination rate constant (K).

C2 =[(S) (F) (Dose/tin) / Cl]*(1-e-ktin)(e-kt2)

Although one would not usually select this equation to demonstrate that the drug concentration is primarily a function of volume of distribution, it is important to recognize that the relationship between the observed drug concentration and volume is not altered as long as the total elapsed time (tin + t2) does not exceed one-third of a half-life.

Our assumption in evaluating the volume of distribution is that although we have not sampled beyond one-third of a t1/2, we have waited until the drug absorption and distribution process is complete.

Clearance

A plasma drug concentration that has been obtained at steady state from a patient who is receiving a constant drug infusion is determined by clearance.

Css ave = (S) (F) (Dose / tau) / Cl

So, the average steady-state plasma concentration is not influenced by volume of distribution. Therefore, plasma concentrations that represent the average steady-state level can be used to estimate a patient's clearnace value, but they cannot be used to estimate a patient's volume of distribution. Generally, all steady-state plasma concentrations within a dosing interval that is short relative to a drug's half-life (tau =<1/3 t1/2) approximate the average concentration. Therefore, these concentrations are also primarily a function of clearance and only minimally influenced by V.

Also the below equation could be used,

Css 1 =[(S)(F)(Dose)/V]/(1-e-kτ)*(e-kt1)

the expected volume of distribution should be retained and the elimination rate constant adjusted such that Css1 at t1 equals the observed drug plasma concentration.

Sensitivity Analysis

Whether a measured drug concentration is a function of clearance or volume of distribution is not always apparent. When this is difficult to ascertain, one can examine the sensitivity or responsiveness of the predicted plasma concentration to a parameter by changing one parameter while holding the other constant. For example, for maintenance infusion, a plasma concentration (C1) at some time intervnal (t1) after a maintenance infusion has been started should be:

C1=[(S)(F)(Dose/τ)/Cl]*(1-e-kt1)

when the fraction of steady that has been reached (1-e-kt1) is small, large changes in clerance are frequently required to adjust a predicted plasma concentration to the appropriate value. If a large percentage change in the clearance value results in a disproportionately small change in the predicted drug level, then something other than clearance is controlling (responsible for) the drug concentration. In this case, the volume of distribution and the amount of drug administered are the primary determinants of the observed concentration. Also in cases where the drug concentration is very low, it might be assay error or sensitivity that is the predominant factor in determining the drug concentration making the ability to revise for any pharmacokinetic parameter limited if not impossible.

This type of sensitivity analysis is useful to reinforce the concept that the most reliable revisions in pharmacokinetic parameters are made when the predicted drug concentration changes by approximately the same percentage as the pharmacokinetic parameter undergoing revision.

When a predicted drug concentration changes in direct proportion, or inverse proportion to an alteration in only one of the pharmacokinetic parameters, it is likely that a measured drug concentration can be used to estimate that patient-specific parameter. But when both clearance and volume of distribution have a significant influence on th prediction of a measured drug concentration, revision of a patient's pharmacokinetic parameters will be less certain because there is an infinite number of combinations for clearance and volume of distribution values that could be used to predict the observed drug concentration. When this occurs, the patient's specific pharmacokinetic characteristics can be estimated by adjusting one or both of the pharmacokinetic parameters. Nevertheless, in most cases additional plasma level sampling will be needed to accurately predict the patient's clearance or volume of distribution so that subsequent dosing regimens can be adjusted.

When the dosing interval is much shorter than the drug's half-life, the changes in concentration within a dosing interval are relatively small, and any drug concentration obtained within a dosing interval can be used as an approximation of the average steady-state concentration. Even though Css max and Css min exist,

Css max =[(S)(F)(Dose)/V]/(1-e-kτ)

and

Css min =[(S)(F)(Dose)/V]/(1-e-kτ)*(e-kτ)

and could be used to predict peak and trough concentrations, a reasonable approximation could also be achieved by using the Css ave, that is

Css ave =(S)(F)(Dose/τ)/Cl

This suggests that even though Css max and Css min do not contain the parameter clearance per se, the elimination rate constant functions in such a way that the clearance derived from Css max or Css min and Css ave would all essentially be the same.

In the situation in which the dosing interval is greater than one-third of a half-life, the use of Css max and Css min are appropriate as not all drug concentrations within the dosing interval can be considered as the Css ave. However, as long as the dosing interval has not been extended beyond one half-life, clearance is still the primary pharmacokinetic parameter that is responsible for the drug concentrations within the dosing interval. Although the elimination rate constant and volume of distribution might be manipulated in Css max and Css min, it is only the product of those two numbers (i.e., clearance) that can be known with any certainty: Cl = (K) (V).

If a drug is administered at a dosing interval that is much longer than the apparent half-life, peak concentrations may be primarily a function of volume of distribution. Since most of the dose is eliminated within a dosing interval, each dose can be thought as something approaching a new loading dose. Of course for steady-state conditions, at some point within the dosing interval, the plasma concentration (Css ave) will be determined by clearance. Trough plasma concentrations in this situation are a function of both clearance and volume of distribution. Since clearance and volume of distribution are critical to the prediction of peak and trough concentrations when the dosing interval is much longer than the drug t1/2, a minimum of two plasma concentrations is needed to accurately establish patient-specific pharmacokinetic parameters and a dosing regimen that will achieve desired peak and trough concentrations.

Choosing A Model to Revise or Estimate A Patient's Clearance at Steady State

As previously discussed, a drug's half-life often determines the pharmacokinetic equation that should be used to make a revised or patient-specific estimate of a pharmacokinetic parameter. A common problem encountered clinically, however, is that the half-life observed in the patient often differs from the expected value. Since a change in either clearance or volume of distribution or both may account for this unexpected value, the pharmacokinetic model is often unclear. One way to approach this dilemma is to first calculate the expected change in plasma drug concentration assocaited with each dose:

delta C = (S) (F) (Dose) / V

where delta C is the change in concentration following the administration of each dose into the patient's volume of distribution. This change in concentration can then be compared to the steady-state trough concentration measured in the patient.

(S) (F) (Dose) / V versus Css min

or

delta C versus Css min

When the dosing interval (tau) is much less than the drug half-life, delta C will be small when compared to Css min. As the dosing interval increases relative to tau, delta C will increase relative to Css min. Therefore, a comparison of delta C or (S) (F) (Dose) / V to Css min can serve as a guide to estimating the drug t1/2 and the most appropriate pharmacokineitc model or technique to use for revision. With few exceptions, drugs that have plasma level monitoring are most often dosed at intervals less than or equal to their half-lives. Therefore, clearance is the pharmacokinetic parameter most often revised or calculated for the patient in question. The following guidelines can be used to select the pharmacokinetic model that is the least complex and therefore the most appropriate to estimate a patient-specific pharmacokientic parameter.

Condition 1

When, (S) (F) (Dose) / V =< 1/4 Css min

Then, tau =<1/3 t1/2

Under these conditions, Css min ≈ Css ave

And Cl can be estimated by Cl = (S) (F) (Dose / tau) / Css ave

Rules/Conditions: Must be at steady state.

Condition 2

When, (S) (F) (Dose) / V =< Css min

Then, tau =< t1/2

Under these conditions, Css min + (1/2) (S) (F) (Dose) / V ≈ Css ave

And Cl can be estimated by Cl = (S) (F) (Dose / tau) / Css ave

Rules/Conditions: Must be at steady state; C is Css min; Bolus model for absorption is acceptable (dosage form is not sustained release; short infusion model is not required, that is, tin =<1/6t1/2)

Conditon 3

When, (S) (F) (Dose) / V > Css min

Then, tau > t1/2

Under these conditions: Css min + (S) (F) (Dose) / V = Css max

where V is an assumed value from the literature.

K is revised (Krevised):

Krevised = ln {[(Css min + (S) (F) (Dose / V)] / Css min} / tau = ln (Css max / Css min) / tau

Rules/Conditions: Must be at steady state; C is Css min; Bolus model for absorption is acceptable (dosage form is not sustained release; short infusion model is not required, that is, tin =< 1/6 t1/2)

Note that the approaches used become more complex as the dosing interval increases relative to the drug half-life. If a drug is administered at a dosing interval less than or equal to one-third of its half-life and the technique in Condition 3 is used to revise clearance, the revised clearance would be correct. The calculation is not wrong, just unnecessarily complex. However, if a drug is administered at a dosing interval that exceeds one half-life and the technique in Condition 1 is used to revise clearance, the revised clearance value would be inaccurate because Css min cannot be assumed to be approximately equal to Css ave. While it could be argued that the technique used in Condition 3 would suffice for all the previous conditions, it is more cumbersome and tends to focus on the intermediate parameters, K and V rather than Cl. One should also be ware that as the dosing interval increases, relative to the drug's half-life, the confidence in a revised clearance diminishes because the volume of distribution, which is an assumed value from the literature, begins to influence the revised clearance to a greater degree. As a general rule, the confidence in Cl is usually good when the dosing interval is < t1/2, steady state has been achieved, and drug concentrations are obtained properly.