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PubMed1998

Statistical Inference (part 1): Basic Concepts.

In the most common research situations, the investigator cannot directly assess the whole population of interest. For this reason, a sample is often studied to infer the actual population measures or parameters. Statistical inference comprises the application of methods to analyze the sample data in order to estimate the population parameters. The basic assumption in statistical inference is that each individual within the population of interest has the same probability of being included in a specific sample. When the sample is not randomly selected. the study findings can still be generalized if the sample can be considered representative of the whole population of interest. A set of statistical methods used to infer the population parameters is performed under the assumption that the sample estimates follow a bell-shaped distribution, called normal distribution. This article presents with the help of examples, the logic used in the sampling distribution theory. The concept of normal (also called gaussian) sampling distribution has an important role in statistical inference, even when the population values are not normally distributed. In fact, in the statistical inference process, the form of the distribution of the sample estimates is more important than the distribution of the individual values.

Journal Article↗

PubMed1983

Immunohistochemical studies of amyloid P component distribution in normal human skin.

The distribution of amyloid P component (AP) in normal human skin was investigated by a light and electron microscopic immunoperoxidase technique, using antibodies to serum amyloid P component (SAP). AP, or an immunologically cross-reactive protein, was found to be specifically localized to the microfibrils of papillary oxytalan fibers and to the peripheral microfibrillar mantle surrounding the elastin core of mature elastic fibers in the reticular dermis; collagen fibers were not stained with anti-SAP. AP was not detected in the dermal-epidermal basement membrane or in the basement membranes surrounding dermal papillary blood vessels and eccrine structures. These findings, which establish the detailed distribution of normal tissue AP in the skin, provide a basis for further studies of the function and behavior of this protein in health and disease.

Amyloid↗

PubMed1997

Log-normality of distribution of occupational exposure concentrations to cobalt.

In evaluating occupational exposures, the daily exposure levels of a worker are generally assumed to be distributed log-normally between days. When there are many workers in a job group, the individual worker's arithmetic means are assumed to be distributed log-normally. Using the cobalt exposure data in a hard metal factory, these assumptions were examined with the Shapiro-Wilk W test. The hypothesis of log-normality was accepted in all cases, while the hypothesis of normality was rejected in some cases.

Cobalt↗

PubMed1978

A comparative study of the buoyant density distribution of normal and malignant lymphocytes.

Density distribution patterns of normal and malignant lymphocytes were compared following centrifugation to equilibrium on linear density gradients. For normal lymphocytes differences in the distribution patterns were observed between: (1) B and T cells, (2) central and peripheral cells, and (3) resting and activated cells. The findings suggested that cell density is determined by cell lineage the degree of differentiation and the stage of functional activation. Marked differences in the density distribution profiles were also observed among certain types of morphologically distinguishable lymphoproliferations. To some extent density analyses enabled the discrimination between CLL, follicular lymphomas and lymphoblastic lymphomas as well as between O-ALL and T-ALL. Density profiles of malignant lymphocytes failed to disclose any features specific for malignancies. But they revealed some similarities with distinct subsets of normal lymphocytes, i.e. between: (1) CLL and bone marrow lymphoid cells, (2) follicular lymphomas and follicular centre cells, and (3) lymphoblastic lymphomas and activated lymphocytes. These findings are further evidence supporting the hypothesis that the malignant transformation of phenotypically different lymphoproliferations takes place at different levels of lymphocyte differentiation.

B-Lymphocytes↗

PubMed1989

Microvascular distribution in normal human synovium.

The distribution of vessels in normal human synovium has been investigated using frozen sections and staining with Ulex Europaeus lectin by an indirect immunoperoxidase technique. The presence of a vascular net close to the tissue surface was confirmed. Quantitative analysis showed the majority of vessel profiles to be small venules with a peak density 25-100 microns beneath the tissue surface. A smaller number of capillaries was present with a peak density within 25 microns of the tissue surface. It is suggested that important functions of the numerous venules seen in synovium may include the provision of a source of fresh monocytes to replace surface macrophages and provision of nutrients for the surface layer of cells as a whole, and that the density of the vascular net may reflect angiogenic signals from this cell layer.

Anthropometry↗

PubMed1983

Intrauterine growth retardation detected in several species by non-normal birthweight distributions.

The statistical distribution of birthweights in 64 litters of pigs, 48 litters of dogs, 59 litters of rabbits, 130 litters of rats and 46 litters of mice was examined. Birthweight followed a Gaussian or normal distribution in about two-thirds of the litters inspected, as judged visually using a normal probability plot, and by the W-test statistic. In the remainder, a single outlying observation in an otherwise normal sample was detected by Dixon's test, and/or a separate sub-population of low birthweight animals could be identified by fitting two normal populations to the observed samples. In the non-Gaussian litters, the average proportion of affected members was 40% of the litter. These results suggest that growth-retarded neonates should not be regarded simply as the tail-end of a normal distribution.

Animals↗

PubMed2004

Consequences of transforming measures of efficacy for acute therapies: 5-HT agonists as a worked example.

OBJECTIVES: To examine whether the distributions of active and placebo response rates change after transformation into therapeutic gains, numbers needed to treat, and therapeutic ratios for 51 published clinical trials of 5-HT(1B/1D) agonists in the acute treatment of migraine. BACKGROUND: Acute migraine therapies have been compared using meta-analysis. Before pooling the results of noncontemporaneous clinical trials for meta-analysis, reconciliation of active response rates with concurrent placebo controls is done because of fluctuations in placebo response rates. The 3 common methods of reconciliation are to find for each clinical trial the therapeutic gain, the number needed to treat, or the therapeutic ratio. METHODS: Raw active and placebo response rates were tabulated and displayed as histograms. The distributions of the therapeutic gains, numbers needed to treat, and therapeutic ratios were similarly plotted. These distributions were then compared with normal distributions using chi-squared goodness-of-fit methodology. RESULTS: The distribution of active response rates was consistent with a normal distribution (passing a goodness-of-fit test). Placebo response rates were not normally distributed. The distribution of therapeutic gains failed a test of normality (P=.018), as did the numbers needed to treat (P <.001); the skews of these distributions were toward opposite ends of the scale for relative efficacy. The therapeutic ratios also failed a test for normality, but passed a test for log-normal distribution. CONCLUSIONS: Choice of transformation is a potential source of bias in meta-analysis of acute migraine therapies.

Acute Disease↗

PubMed1988

Assessment of polyclonal antibody binding of ligand by Sips' equation or by the exact polyclonal equation. Comparison of models.

Different possibilities of modelling equilibrium data on polyclonal antibody binding of ligand were investigated. The binding curves of the exact polyclonal model, based upon the mass-action law, and the Sips' model were compared. The basic assumption was that the free energy of binding is distributed according to the Gauss normal distribution. Using a few equations, the values of binding site concns and affinity constants of the individual clones were converted to the parameters of Sips' equation (that is, A = Ab binding site concn, K0 = average affinity, Hi = heterogeneity index). It was demonstrated that the binding curves from the exact polyclonal and the Sips' equations were very similar for simulated, approximately normally distributed antibody populations. Even when only two or three Ab-clones are involved, the Sips' and the exact polyclonal binding curves are very similar as long as the ratio between affinity constants does not exceed 10. Equilibrium data plotted as bound ligand concn on a linear scale vs free ligand concn on a logarithmic scale (or alternatively after logarithmic transformation of free ligand concn) form the well known sigmoid saturation curve. A mathematical relationship was demonstrated between half-height slope of such curves, resulting from approximately normally distributed antibody populations, and parameters of Sips' equation. The half-height slope is Hi X A/4 when natural logarithms are used. The ideas described were illustrated by weighted nonlinear curve-fittings applied to actual equilibrium data on anti-DNP antibodies. Tested in this way, the Sips' and a 2-clonal model gave equally good fit but somewhat different average affinities. It was concluded that it is often impossible to distinguish between a 2-clonal, a 3-clonal or an approximately normally distributed antibody population by curve-fitting to experimental equilibrium data.

2,4-Dinitrophenol↗

PubMed2002

Assessment of type I error rates for the statistical sub-model in NONMEM.

The aim of this study was to assess the type I error rate when applying the likelihood ratio (LR) test, for components of the statistical sub-model in NONMEM. Data were simulated from a pharmacokinetic one compartment intravenous bolus model. Two models were fitted to the data, the simulation model and a model containing one additional parameter, and the difference in objective function values between models was calculated. The additional parameter was either (i) a covariate effect on the interindividual variability in CL or V, (ii) a covariate effect on the residual error variability, (iii) a covariance term between CL and V, or (iv) interindividual variability in V. Factors in the simulation conditions (number of individuals and samples per individual, interindividual and residual error magnitude, residual error model) were varied systematically to assess their potential influence on the type I error rate. Different estimation methods within NONMEM were tried. When the first-order conditional estimation method with interaction (FOCE INTER) was used the estimated type I error rates for inclusion of a covariate effect (i) on the interindividual variability, or (ii) on the residual error variability, were in agreement with the type I error rate expected under the assumption that the model approximations made by the estimation method are negligible. When the residual error variability was increased, the type I error rates for (iii) inclusion of covariance between etaCL-etaV were inflated if the underlying residual distribution was lognormal, or if a normal distribution was combined with too little information in the data (too few samples per subject or sampling at uninformative time-points). For inclusion of (iv) etaV, the type I error rates were affected by the underlying residual error distribution; with a normal distribution the estimated type I error rates were close to the expected, while if a non-normal distribution was used the type I errors rates increased with increasing residual variability. When the first-order (FO) estimation method was used the estimated type I error rates were higher than the expected in most situations. For the FOCE INTER method, but not the FO method, the LR test is appropriate when the underlying assumptions of normality of residuals, and of enough information in the data, hold true. Deviations from these assumptions may lead to inflated type I error rates.

Analysis of Variance↗

PubMed1987

Secretory ameloblasts and calcium distribution during normal and experimentally altered mineralization.

The distribution of calcium in relation to secretory ameloblasts of the rat incisor was studied. An experimental model system in which enamel mineralization was temporarily inhibited by injecting sodium fluoride and cobalt chloride was used. Potassium pyroantimonate (PPA) cytochemistry, electron energy loss spectroscopy (EELS), and energy dispersive X-ray spectrometry (EDS) were used to clarify the role of the ameloblast in controlling calcium distribution during normal and experimentally altered enamel mineralization. Secretory ameloblasts chemically-preserved in glutaraldehyde either with or without PPA were analyzed for calcium; those preserved with PPA showed higher concentrations of calcium than did those preserved with glutaraldehyde only. Freeze-dried control and experimental tissues showed an increasing gradient of calcium from stratum intermedium cells to the distal ends of the ameloblasts. Calcium levels were reduced near the distal ends of the cells following fluoride and cobalt injections, while magnesium levels were increased markedly in the same region. This multi-method approach showed correlated calcium localization in specific regions of this cell in relation to changes in function. The study thus provides additional evidence for active involvement of the ameloblasts in enamel mineralization.

Ameloblasts↗

PubMed1995

Lymphocyte phenotype and subset distribution in normal cerebrospinal fluid.

The distribution of lymphocyte subpopulations in cerebrospinal fluid (CSF) and their phenotypic characteristics were extensively investigated in a group of 18 healthy individuals using two- and three-color flow cytometry. Generally, CD3+ T lymphocytes constituted the vast majority of CSF lymphocytes while the number of B lymphocytes and NK cells were low. Most T lymphocytes exhibited the phenotype of memory/primed cells in both the CD4+ and CD8+ subpopulations. Two markers for recent activation, HLA-DR and interleukin-2 receptor (CD25) were not upregulated when compared with peripheral blood (PB) in the majority of CSF T lymphocytes. However, a fraction of T lymphocytes co-expressing the NK cells markers CD56 and/or CD16 showed a pronounced upregulation of HLA-DR in CSF as compared with PB. This study documents that the cellular composition of the normal CSF differs profoundly from PB regarding all major lymphocyte subpopulations. This has to be taken into account in studies addressing questions regarding cellular immune reactions in the central nervous system under pathological conditions.

Adult↗

PubMedDate not supplied

Evaluation of sampling plans to detect Cry9C protein in corn flour and meal.

StarLink is a genetically modified corn that produces an insecticidal protein, Cry9C. Studies were conducted to determine the variability and Cry9C distribution among sample test results when Cry9C protein was estimated in a bulk lot of corn flour and meal. Emphasis was placed on measuring sampling and analytical variances associated with each step of the test procedure used to measure Cry9C in corn flour and meal. Two commercially available enzyme-linked immunosorbent assay kits were used: one for the determination of Cry9C protein concentration and the other for % StarLink seed. The sampling and analytical variances associated with each step of the Cry9C test procedures were determined for flour and meal. Variances were found to be functions of Cry9C concentration, and regression equations were developed to describe the relationships. Because of the larger particle size, sampling variability associated with cornmeal was about double that for corn flour. For cornmeal, the sampling variance accounted for 92.6% of the total testing variability. The observed sampling and analytical distributions were compared with the Normal distribution. In almost all comparisons, the null hypothesis that the Cry9C protein values were sampled from a Normal distribution could not be rejected at 95% confidence limits. The Normal distribution and the variance estimates were used to evaluate the performance of several Cry9C protein sampling plans for corn flour and meal. Operating characteristic curves were developed and used to demonstrate the effect of increasing sample size on reducing false positives (seller's risk) and false negatives (buyer's risk).

Algorithms↗

PubMed1984

Estimation of response slopes in respiratory control.

The ventilatory response to changes in alveolar (arterial) CO2 is widely used as an index of respiratory control behavior. Methods for estimating these response slopes should incorporate the possibility that there may be errors in both the independent (partial pressure of CO2) and dependent (ventilation) variables. In a recent paper Daubenspeck and Ogden (J. Appl. Physiol. Respirat. Environ. Exercise Physiol. 45:823-829, 1978) have suggested problems inherent in the traditional technique of reduced major axis and have suggested a more contemporary technique of directional statistics. We have previously analyzed both techniques and developed a method to overcome the problems of reduced major axis and problems inherent in the use of directional statistics. Under the assumption of a bivariate normal distribution, we demonstrate that our slope estimate is similar to the maximum likelihood estimate proposed by Mardia et al. (J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 54: 309-313, 1983) for this problem. In addition, we demonstrate a bootstrap statistical approach when the distributions are not normally distributed. These concepts are illustrated using O2-CO2 interaction data.

Carbon Dioxide↗

PubMed2006

A bivariate mixture model analysis of body weight and ascites traits in broilers.

The objective of the present study was to use bivariate mixture models to study the relationships between body weight (BW) and ascites indicator traits. Existing data were used from an experiment in which birds were housed in 2 groups under different climate conditions. In the first group, BW, the ratio of right ventricular weight to total ventricular weight (RV:TV), and hematocrit value (HCT) were measured in 4,202 broilers under cold conditions; in the second group, the same traits were measured in 795 birds under normal temperature conditions. Cold-stress conditions were applied to identify individuals that were susceptible to ascites. The RV:TV and HCT were approximately normally distributed under normal temperature conditions, whereas the distributions of these traits were skewed under cold temperature conditions, suggesting different underlying distributions. Fitting a bivariate mixture model to the observations showed that there was only one homogeneous population for ascites traits under normal temperature conditions, whereas there was a mixture of (2) distributions under cold conditions. One distribution contained nonascitic birds and the other distribution contained ascitic birds. In the distribution of nonascitic birds, the inferred phenotypic correlations (phenotypic correlations with 2 distinguishing underlying distributions) of BW with RV:TV and HCT were close to zero (0.10 and -0.07, respectively), whereas in the distribution of ascitic birds, the inferred phenotypic correlations of BW with RV:TV and HCT were negative (-0.39 and -0.4, respectively). The negative inferred correlations of BW with RV:TV and HCT in the distribution of ascitic birds resulted in negative overall correlations (correlations without 2 distinguishing distributions) of BW with RV:TV (-0.30) and HCT (-0.37) under cold conditions. The present results indicate that the overall correlations between BW and ascites traits are dependent on the relative frequency of ascitic and nonascitic birds in the population.

Animals↗

PubMed2004

A quantitative trait locus mixture model that avoids spurious LOD score peaks.

In standard interval mapping of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. At any given location in the genome, the evidence of a putative QTL is measured by the likelihood ratio of the mixture model compared to a single normal distribution (the LOD score). This approach can occasionally produce spurious LOD score peaks in regions of low genotype information (e.g., widely spaced markers), especially if the phenotype distribution deviates markedly from a normal distribution. Such peaks are not indicative of a QTL effect; rather, they are caused by the fact that a mixture of normals always produces a better fit than a single normal distribution. In this study, a mixture model for QTL mapping that avoids the problems of such spurious LOD score peaks is presented.

Genetic Techniques↗

PubMed1994

Ribosome distribution in normal and infarcted rat hearts.

Distribution of ribosomes throughout the myocardium of normal and infarcted rat hearts was studied by immunofluorescence and laser confocal scanning microscopy. In addition, sections were labelled with peroxidase or immunogold particles for electron microscopic examination. Ligation of the proximal free left coronary artery produced severe myocardial ischaemia, and after 6 days of ligation most of the left ventricular wall was necrotic and partially replaced by granulation tissue. Immunofluorescence microscopy revealed the presence of ribosomes throughout the non-necrotic myocardium. Some cardiac muscle cells located in subendocardial areas and in the border areas surrounding the infarct were particularly intensely stained. Cells constituting the granulation tissue frequently exhibited strong ribosomal immunostaining. Within longitudinally sectioned cardiac muscle cells, ribosomes were organized in strands oriented along the long axis of the cell as well as in a cross-striated pattern. By double labelling of muscle cells with antibodies against ribosomes and Z-line-associated proteins (desmin or alpha-actinin), it was shown that the cross-striated bands of anti-ribosomal staining coincided with the I-bands along the myofibrils. Immunoelectron microscopy confirmed a wide distribution of ribosomes throughout the intermyofibrillar and subsarcolemmal sarcoplasm, and some labelling was also observed within the I-band. The present results indicate that ribosomes are distributed in a characteristic pattern throughout the sarcoplasm of cardiac muscle cells in association with the myofibrils. Furthermore, it is suggested that within viable cardiac muscle cells located adjacent to the infarct, protein synthesis is increased; this might be an important factor in regional development of compensatory hypertrophy of the surviving cardiac muscle cells.

Animals↗

PubMed1995

Estimation of quantitative genetic parameters under non-normal models.

In traditional quantitative genetics, the relationship between the observed value of a quantitative trait in a set of families and its genetic and environmental contributions can be described as a linear additive relationship. Generally, it is assumed that the genetic and environmental effects are independent and normally distributed. Consequently, the quantitative trait also has a normal distribution. However, there are some situations where the phenotype does not follow the normal distribution. To deal with this problem the author suggests the families of distributions that belong to the Johnson Translation System (JTS). As an example, two dependent quantitative traits, weight and height, are investigated assuming that they have a lognormal and normal distribution, respectively. Computational methods for estimating the genotypic variances and covariances are presented.

Finland↗

PubMed1994

A quantitative method for cost reimbursement and length of stay quality assurance in multiple trauma patients.

OBJECTIVE: To develop a statistically valid method for trauma reimbursement and quality assurance (QA) length-of-stay filters. This is needed because diagnosis related group (DRG)-based trauma payment systems assume a random sampling of injury severities from a normally distributed population and thus result in economic disincentives to level I trauma centers. METHODS: 142 trauma patients with MVC blunt multisystem injuries (MSI) (ISS > or = 16) were studied concurrently during their hospital course. SETTING: Level I regional trauma center. OUTCOME MEASURES: Outcome measures were (dependent variables) length of stay (LOS) and state-approved hospital charges (COST). RESULTS: Mean acute care COST was $74,310, but the distribution of COST was log normal, rather than Gaussian normal as assumed by DRGs. The LOS for MSI was more than twice the average for all trauma (22 vs. 9 days), reflecting skewed severities of level I patients and was related to COST (r2 = 0.802; p < 0.0001). The ISS alone was a weak determinant of COST or LOS (r2 = 0.05; p < 0.0001). The best single determinant of COST and LOS was survival (r2 = 0.15; p < 0.0001): as it increased, it increased LOS. The most costly injuries (all p < 0.0001) involved the lower extremity (LE) or hip joint (HIP), whereas sepsis and pulmonary and surgical complications constituted the most costly complications (all p < 0.0001). Regression models that accounted for the log-normal distribution of the dependent variable and based on binary variables for survival, LE and HIP injuries, and the complications of sepsis, ARDS, pulmonary failure, MOFS, plus ISS, explained nearly two thirds of the variability in COST (r2 = 0.621; p < 0.0001) or LOS (r2 = 0.687; p < 0.0001) and the residuals were normally distributed. CONCLUSIONS: These models provide a valid method of reimbursement for MSI trauma for level I trauma centers, since the data imply that good care associated with survival from specific complications of MSI are the major determinants of COST, rather than the specific type of injury or the resultant ISS. Moreover, using survival and ISS plus the disease-related complications as determinants of LOS, this method can be applied to any U.S. region since local factors can be used to adjust hospital COST as a highly correlated function of LOS. This method also permits identification of LOS outliers for QA, taking into account the influence of injury complications.

Accidents, Traffic↗