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Mark has done a great job in writing complex statistical concepts in an easy to understand format that makes grasping them both easy to understand and to use.
With the help of Mark's book, and some diligent studying, I received an A in my stats course. Excel Statistical Master that taught me all of the basic concepts for the different tests we used. Unlike the indecipherable jargon in the countless books I have wasted money on, the language in this book is plain and easy to understand.
This same problem above is solved in the Excel Statistical Master with only 3 Excel formulas (and NO looking anything up on a Z Chart).
This same problem above is solved in the Excel Statistical Master with only 3 Excel formulas (and not having to look anything up on a Z Chart). This same problem above is solved in the Excel Statistical Master with only 1 Quick Excel formula (and NO looking anything up on a Z Chart).
This same problem above is solved in the Excel Statistical Master with only 2 Excel formulas (and not having to look anything up on a Z Chart). This would state that the actual population Proportion has a 95%probability of lying within the calculated interval.
One approach for validating a new analytical method is to analyze a sample containing a known amount of analyte, μ.
Figure 4.14 Relationship between confidence intervals and the result of a significance test. Using α = 0.05, is there any evidence that the analysis is giving inaccurate results?
Because there is no reason to believe that the results for the standard must be larger (or smaller) than μ, a two-tailed t-test is appropriate. Is there any evidence at α = 0.05 that there is a determinate error affecting the results? Earlier we made the point that you need to exercise caution when interpreting the results of a statistical analysis.
If we regularly analyze a particular sample, we may be able to establish the expected variance, σ2, for the analysis. We can use an F-test to evaluate whether a difference between s2 and σ2 is significant.
Is there any evidence at α = 0.05 that there is a significant difference in the variance between the results for these two samples? Three factors influence the result of an analysis: the method, the sample, and the analyst.
Before we consider the significance tests for comparing the means of two samples, we need to make a distinction between unpaired data and paired data.
One simple test for determining whether data are paired or unpaired is to look at the size of each sample.
Consider two analyses, A and B with means of XA and XB, and standard deviations of sA and sB.
Thus far our development of this t-test is similar to that for comparing X to μ, and yet we do not have enough information to evaluate the t-test. To determine whether we can use a pooled standard deviation, we first complete an F-test of the following null and alternative hypotheses.
To compare the means for the two analysts we use the following null and alternative hypotheses. Suppose we are evaluating a new method for monitoring blood glucose concentrations in patients. When using paired data we first calculate the difference, di, between the paired values for each sample. Acquiring samples over an extended period of time introduces a substantial time-dependent change in the concentration of monensin. Suppose you are studying the distribution of zinc in a lake and want to know if there is a significant difference between the concentration of Zn2+ at the sediment-water interface and its concentration at the air-water interface. One important requirement for a paired t-test is that determinate and indeterminate errors affecting the analysis must be independent of the analyte's concentration. Earlier in the chapter we examined several data sets consisting of the mass of a circulating United States penny.
Figure 4.15 Dotplots showing the distribution of two data sets containing a possible outlier. Our New BMJ website does not support IE6 please upgrade your browser to the latest version or use alternative browsers suggested below.
Fig 4 Random effects model (adjusted for various confounding factors) showing risk of stillbirth associated with previous stillbirth (confounders vary between studies).
Subgroup analysisBecause of methodological differences between studies that examined risk of recurrent unexplained stillbirth we were unable to perform the prespecified subgroup analysis. DiscussionIn this systematic review and meta-analysis, women who experienced a stillbirth in an initial pregnancy experienced nearly a fivefold increase in the odds of stillbirth in a subsequent pregnancy.
You'll be able to grasp your statistics course a LOT easier with the Excel Statistical Master. If you found your statistics book confusing, You'll really like the Excel Statistical Master. Because the area between any two limits of a normal distribution is well defined, constructing and evaluating significance tests is straightforward. Student was the pen name for William Gossett (1876-1927) who developed the t-test while working as a statistician for the Guiness Brewery in Dublin, Ireland. Since texp is greater than t(0.05,4) we reject the null hypothesis and accept the alternative hypothesis. This often is the case, for example, in clinical labs that routinely analyze hundreds of blood samples each day. We can study the influence of these factors by conducting experiments in which we change one of the factors while holding the others constant. This is a critical distinction and learning to distinguish between the two types of data is important.
This is an example of paired data because we use the same 10 pennies to evaluate each balance. Weigh the pennies in the first group on one balance and weigh the second group of pennies on the other balance.
In this case we can replace sA2 and sB2 with a pooled variance, (spool)2, that provides a better estimate for the variance. Using these difference values, we then calculate the average difference, d, and the standard deviation of the differences, sd.
Because the variation in concentration between samples is so large, we use paired t-test with the following null and alternative hypotheses.

If this is not the case, then a sample with an unusually high concentration of analyte will have an unusually large di. The null hypothesis is that there are no outliers, and the alternative hypothesis is that there is an outlier. To protect against rejecting a valid data point, we usually apply the more conservative two-tailed Q-test, even though the possible outlier is the smallest or the largest value in the data set. If Gexp is greater than G(α,n), we may reject the data point as an outlier, otherwise we retain the data point as part of the sample. Supplementary efforts included searching relevant internet resources as well as hand searching the reference lists of included studies. Even when restricting the analysis to first and second pregnancies, the risk of stillbirth in the second pregnancy was increased if the first pregnancy ended in stillbirth.
He published under the name Student because the brewery did not want its competitors to know they were using statistics to help improve the quality of their products. At the 95% confidence level the difference between X and μ is too large to be explained by indeterminate sources of error, suggesting that a determinate source of error is affecting the analysis.
Having determined that a result is inaccurate, as we did in Example 4.16, the next step is to identify and to correct the error.
A few replicate analyses of a single sample gives a sample variance, s2, whose value may or may not differ significantly from σ2.
One explanation for the difference might be that the aspirin tablets were not selected randomly.
For example, to compare two analytical methods we can have the same analyst apply each method to the same sample, and then examine the resulting means. Here are two simple examples that highlight the difference between unpaired data and paired data. We will learn why this distinction is important when we review the significance test for paired data; first, however, we present the significance test for unpaired data. Including this sample in the calculation of d and sd leads to a biased estimate of the expected mean and standard deviation. Regardless of its source, the presence of an outlier compromises any meaningful analysis of our data. The Q-test compares the gap between the suspected outlier and its nearest numerical neighbor to the range of the entire data set (Figure 4.15). If Qexp is greater than Q(α, n), then we reject the null hypothesis and may exclude the outlier. Where published information was unclear or inadequate, corresponding authors were contacted for more information.Study selection Cohort and case-control studies from high income countries were potentially eligible if they investigated the association between stillbirth in an initial pregnancy and risk of stillbirth in a subsequent pregnancy. In the meta-analysis using adjusted odds ratios from primary studies the increased odds of recurrence remained, although it was slightly less than the unadjusted analysis. Although we can reject the null hypothesis at the 98% confidence level, we cannot reject it at the 99% confidence level.
Before expending time and money on this, however, you should first critically examine your data. This is an example of unpaired data because we evaluate each balance using a different sample of pennies. Because the variation in the blood glucose levels amongst patients is large we may be unable to detect a small, but significant difference between the methods if we use different patients to gather data for each method.
Samples were collected from the fermentation vats at various times during production and analyzed for the concentration of monensin using both methods.
There are many significance tests for identifying potential outliers, three of which we present here.
Stillbirth was defined as fetal death occurring at more than 20 weeks’ gestation or a birth weight of at least 400 g. Twelve studies reported adjusted odds ratios for the association between stillbirth in an initial pregnancy and risk of stillbirth in a subsequent pregnancy.
Where the primary studies had specifically looked at unexplained stillbirth, the evidence was less clear cut. For (b) we must reject the null hypothesis because there are portions of the sample’s confidence interval that lie outside the confidence interval due to indeterminate error.
Using paired data, in which the we analyze each patient’s blood using both methods, prevents a large variance within a population from adversely affecting a t-test of means. When paired data span a wide range of concentrations, however, the magnitude of the determinate and indeterminate sources of error may not be independent of the analyte's concentration. Two reviewers independently screened titles to identify eligible studies based on inclusion and exclusion criteria agreed a priori, extracted data, and assessed the methodological quality using scoring criteria from the critical appraisal skills programme. As data overlapped slightly (as little as 8%) between the studies by Black and colleagues17 and Bhattacharya and colleagues,14 we conducted a sensitivity analysis by removing Black and colleagues’ study—the rationale being that Bhattacharya and colleagues’ findings were considered more generalisable because of the population based design. Only two studies had looked at unexplained stillbirth in an initial pregnancy and any stillbirth (explained or unexplained) in the subsequent pregnancy and had found no increased risk. In the case of (c) we retain the null hypothesis because the confidence interval due to indeterminate error completely encompasses the sample’s confidence interval. If the standard deviation for your analysis is unrealistically small, the probability of a type 2 error increases. In such cases a paired t-test may give misleading results since the paired data with the largest absolute determinate and indeterminate errors will dominate d. We might ask whether this penny’s mass is so different from the other pennies that it is in error.
However, two other studies had specifically assessed unexplained stillbirth after any stillbirth (explained or unexplained) and reported a greatly increased risk of recurrence.Strengths and limitations of this reviewThis systematic review and meta-analyses offers the first comprehensive synthesis of the available evidence on the association between stillbirth and unexplained stillbirth in a previous pregnancy and risk of recurrence. Including a few additional replicate analyses of the standard and reevaluating the t-test may strengthen your evidence for a determinate error, or it may show that there is no evidence for a determinate error. In this situation a regression analysis, which is the subject of the next chapter, is more appropriate method for comparing the data. Subgroup analysis was performed on studies that examined unexplained stillbirth.Results 13 cohort studies and three case-control studies met the inclusion criteria and were included in the meta-analysis.
Studies were published between 2001 and 2014 and five were conducted in Australia,18 38 40 41 42 three in Scotland,14 17 39 three in the United States,15 33 44 and one each in Denmark,35 Israel,43 the Netherlands,37 Norway,36 and Sweden.34 All articles were in English. Meta-analyses were conducted (unadjusted and adjusted) that included data on a large number of women from high income countries to provide a quantitative summary of the results.
Nine of the cohort studies were large population based studies that included data extending over at least 10 years. The population based design of the included studies is a strength that promotes generalisability within countries as well as transferability of findings to other high income countries. One of these36 included data collected over nearly 40 years, thus the combined data collection period spanned from 1967 to 2009. Statistical heterogeneity of studies was substantial as evidenced by the high I2 statistic, and therefore as with all reviews of observational studies the findings should be interpreted with caution. Eleven of the cohort studies examined the risk of stillbirth recurrence in a second pregnancy.

Nevertheless, in all comparisons the estimates showed the same direction of effect, which suggests that the association is real.46Systematic reviews of observational studies typically combine studies that are diverse both clinically and methodologically. In one of the remaining cohort studies, data were available on a subset of women on their first and second sequential births.40 The remaining cohort study examined risk of recurrence of unexplained stillbirth and included women in the exposed group with an unexplained stillbirth that need not necessarily have been their first birth.
The supplementary file provides details of all sensitivity analyses.Population attributable riskWe calculated the population attributable risk percentage to assess the proportion of subsequent stillbirth that is attributable to stillbirth in a first pregnancy.
As a result, heterogeneity between the results is to be expected.46 In our analyses, primary studies differed in their definition of stillbirth and in their use of classification systems for determining cause of death and consequently in the classification of unexplained stillbirth.
Moreover, methodological differences were apparent in their lack of consistency in tackling the effects of confounding. For instance, some studies adjusted for causal factors such as pre-eclampsia and placental abruption or preterm birth, which is not a confounder but rather in the causal path of stillbirth. Methodological differences between these studies precluded pooling the results.Conclusions The risk of stillbirth in subsequent pregnancies is higher in women who experience a stillbirth in their first pregnancy. After adjusting only for maternal characteristics, the estimated risk from these studies was reduced by 22% compared with the unadjusted risk. This suggests that much of the risk of recurrence is not explained by modifiable maternal factors, consistent with the Stillbirth Collaborative Research Network study,44 which showed that apart from the occurrence of previous stillbirth or pregnancy loss, risk factors known at confirmation of pregnancy (a combination of demographic and obstetric maternal characteristics) explained only a small amount of the risk of stillbirth. Evidence surrounding the recurrence risk of unexplained stillbirth remains controversial.IntroductionOver the past two decades many high income countries have achieved substantial reductions in late gestation stillbirths. Norway and the Netherlands show the largest reductions; however, in the United Kingdom the downward trend in stillbirth rates has slowed and become more or less stable.
Residual confounding from poor measurement of these could still explain at least part of the associations reported.
The literature on stillbirth has recently expanded3; however, studies that examined the recurrence of stillbirth remain scarce. Some studies report recurrence risks ranging from twofold to 10-fold,14 15 16 whereas others report no increased risk.17 18 Although stillbirth is a common obstetric complication its recurrence is rare and it may be that some primary studies lack the power to detect any increase in risk.
Primary reports in the literature investigating the risk of stillbirth recurrence yielded inconsistent results, but most published studies suggest an increased risk for women with a history of stillbirth. Furthermore, many causes of stillbirth (for example, placental abruption) are known to recur in subsequent pregnancies, thus increasing the chances of another stillbirth associated with that cause; but in cases where stillbirth remains unexplained there is no consensus about the risk of stillbirth in the next pregnancy.
However, when the previous stillbirth has been unexplained or when the sample size was small, adjustment for confounding factors made confidence intervals cross unity and no increased risk in subsequent stillbirth was found.Inconsistency in the definition of unexplained stillbirth has also been recognised and it has been pointed out that truly unexplained stillbirths are those in which no cause of death can be found despite thorough postmortem examination. Because of the uncertainty surrounding the recurrent risk for stillbirth it is difficult for clinicians to counsel couples and to know what level of care to provide in subsequent pregnancies.We reviewed the evidence on the association between stillbirth in an initial pregnancy and risk of stillbirth in subsequent pregnancies.
Gordon and colleagues made the decision a priori to only analyse data from 2002 because from that point all deaths were routinely classified using the perinatal death classification system of the Perinatal Society of Australia and New Zealand. Specifically, we hypothesised that women whose first pregnancy resulted in a stillbirth or an unexplained stillbirth had an increased risk of stillbirth in any subsequent pregnancy compared with women who had a previous live birth. This stillbirth classification system incorporates policy directives that include recommendations to discuss and offer postmortem examinations to every affected family, and for examination of the placenta.
Even so, a postmortem examination was performed in only half of the stillbirths in the cohort.
The authors draw attention to the low rate of postmortem examinations undertaken in unexplained stillbirths in New South Wales during the study period (30.8%).
During preliminary searches we found that the two concepts of stillbirth and recurrence were more often included in journal abstracts or indexed.
Rates of postmortem examination for which parental consent is required are also low in the UK (around 45%), although in Scotland as a result of ongoing commitment to improve procedure, rates of consent are higher.Nijkamp and colleagues55 evaluated the subsequent pregnancy outcome after a previous stillbirth.
The cause of death in both the index and the subsequent pregnancy was determined and compared using the Tulip classification system, the system developed and currently in use in the Netherlands for classifying cause of death. We used a combination of Medical Subject Headings key words, and text words for “stillbirth”, “recurrence”, “pregnancy”, and “risk factors” that appeared in abstracts and titles. Of 163 women, 11 had a subsequent stillbirth, and of these at least six showed an association between the cause of death in both events.Stillbirth is a relatively uncommon outcome in high income countries and recurrence even more so. No restrictions were applied to date, publication, or language, although we limited studies to those in human participants. Therefore to provide statistical power to observe the recurrence of unexplained stillbirth, large numbers of births are necessary in primary studies. Systematic reviews and meta-analyses can help overcome this deficiency in primary analyses.Implications for clinical practice and policyThis research is relevant to public health and clinical practice because it adds to the body of evidence on stillbirth recurrence and can be used to counsel couples who are thinking of conceiving after a previous explained or unexplained pregnancy loss. The search strategy was initially developed for use in Medline and was then adapted for searching the other databases (see supplementary file for the search strategies used in each database). Smoking and obesity are independently associated with an increased risk of stillbirth, and modification of these lifestyle factors may make a small but important reduction in the risk of recurrence.
Current management of pregnancies should take account of pregnancy history and make use of prepregnancy counselling services.
Based on the available evidence identified by this review, a stillbirth in an initial pregnancy was associated with an increased risk of a subsequent stillbirth, and pregnancies after a stillbirth should be closely monitored with a view to intervene at the first sign of fetal compromise. Consequently, clinical guidance from the UK Royal College of Obstetricians and Gynaecologists recommends that pregnant women with a previous stillbirth should be managed as high risk, yet many stillbirths result from non-recurrent events such as infection, problems with the umbilical cord, and isolated structural fetal anomalies. This was done using the generic inverse weighted method—that is, studies were weighted by the inverse of the standard error of the log transformed odds ratios.30 We calculated the standard errors of these log odds ratios using published confidence intervals and then used these to weight the studies according to the precision of the odds ratio.
There is little evidence that this approach actually prevents stillbirth in the next pregnancy without increasing morbidity from unnecessary interventions.The demand for international consensus on the use of a universal definition and classification for stillbirth for research purposes has been proposed for some time. To explore the definition of stillbirth as a potential source of heterogeneity we conducted a sensitivity analysis.Statistical heterogeneity was assessed using the Cochran’s ?2 test, and the I? statistic used to summarise the degree of variation across studies. To improve understanding of cause related and unexplained risk of stillbirth recurrence, large scale individual patient data meta-analyses are warranted, where uniform definitions and classifications can be applied.
As recommended by the Cochrane handbook for systematic reviews,30 we considered an I? value of 0-40% to represent low heterogeneity, 30-60% moderate heterogeneity, 50-90%, substantial heterogeneity, and 75-100% considerable heterogeneity.Assuming there is a causal relation between a risk factor and a disease, the population attributable risk is the proportion of disease or deaths in a population that can be attributed to an exposure.
This systematic review highlights the scarcity of studies that examined the risk of stillbirth recurrence and shows the need for high quality multicentre studies using standardised definitions of stillbirth and unexplained stillbirth to add to current knowledge.
Future research that stratifies women for the key confounding variables of obesity and smoking is needed to assess the impact of lifestyle modification on risk of recurrent stillbirth. In addition, one study and a conference abstract were identified through supplementary searches (fig 1?). In addition, to ascertain cause related recurrence, population based studies that examine the risk of subsequent stillbirth based on the initial cause of death are also needed.
A clearly standardised universal definition of stillbirth for research and reporting practices is key issue if the methodological quality of stillbirth research is to be improved, be more comparable, and have more impact. Twenty two citations from these did not meet the inclusion criteria and thus were excluded.
Thirteen cohort studies14 15 17 18 33 34 35 36 37 38 39 40 41 and three case-control studies42 43 44 met the inclusion criteria.

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