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Analysis of variance (ANOVA) is a group of tests to compare means of
normally distributed variables between two or more groups (Figure 16.8). Oneway ANOVA is used when we are interested in effect of one factor with two or
more levels, such as three independent groups in a trial. Two-way ANOVA can
handle two factors. For example, a 2 × 2 factorial study could assess the effect
of two factors simultaneously, such as two surgical techniques and two
postoperative protocols in one study (four groups). Other ANOVA tests include
analysis of covariance (ANCOVA), repeated measures ANOVA (RANOVA),
and multivariate ANOVA (MANOVA). ANCOVA provides inclusion of a
continuous covariate such as the baseline value of the outcome. RANOVA is
used when one subject is measured several times during the study, and
MANOVA assesses multiple correlated outcomes simultaneously.
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FIGURE 16.8. Three groups with means of 4, 5, and 6 and SD of 1
(n = 20 in each group). In ANOVA, the F-value is calculated as
(between-groups variance)/(within-group variance). In this example,
the F-value is 20 (calculation not shown) indicating that betweengroup variation is 20 times larger than within-group variation.
Observing such an extreme F-value with a sample size of n = 60 is
unlikely, and the associated P-value is small (<.001). A low P-value
suggests that at least one of the group means is different from the
others. However, the analysis does not specify which one. After
rejecting the null hypothesis, the differences between individual
groups can be tested with post hoc analysis, such as Tukey method,
which adjusts the P-value according to the number of comparisons.
Nonparametric Tests
Nonparametric tests are used in situations when the continuous outcome has a
skewed distribution or with categorical ordinal outcomes. They should be
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employed when the distribution is unclear, and an assumption of normality
cannot be made. Nonparametric tests are based on the ranks, rather than the
actual values of the variables. For example, in the Mann-Whitney U test
(comparing two independent groups), all participants receive a rank based on
the value of the variable to be compared. The sum of ranks, and not the actual
values, is used to determine if we should reject the null hypothesis. The
disadvantages of nonparametric tests are that they only test the null
hypothesis (ie, they return a P-value) and do not estimate the actual effect
size. Another issue with nonparametric tests is that they have lower power
compared with parametric tests. Nonparametric tests also assume that the
groups being compared have similar distributions, an assumption that may be
difficult to meet.
Fisher and Chi-Square Tests
The Fisher exact test and chi-square test are commonly used to determine
whether two categorical variables are associated. For instance, we could
compare the number of adverse events (a binary categorical variable) across
two or more groups (a categorical variable). The chi-square test is appropriate
for larger samples, while the Fisher exact test is also suitable for small
samples.
Correlation Analysis
Correlation analysis can be used to assess the relationship between two
continuous variables, such as age and proximal interphalangeal joint (PIPJ)
ROM. Each patient in a cohort will contribute a pair of outcomes (age; ROM),
and the data from each patient can be plotted on a two-way scatter plot (Figure
16.9). A direct inspection of the scatter plot may show an association.
Mathematically, this relationship can be quantified using the Pearson
correlation coefficient, r, which takes values between −1 and 1.9 A coefficient of
0 indicates no correlation at all, and 1 indicates a perfect linear correlation. The
positive and negative signs indicate proportional and inversely proportional
relationships, respectively. A strong correlation between age and ROM only
implies a linear association and does not establish a causal relationship.
Finally, these tests examine linear relationships only and do not provide
information on nonlinear relationships, such as a U-shaped one. Just like other
parametric tests, the Pearson correlation coefficient test is sensitive to outliers,
and the Spearman rank correlation coefficient (r s ) is more appropriate when
the underlying distribution is not normal.
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FIGURE 16.9 Two hypothetical correlations between age and range
of motion (ROM). In the plot on the left, the correlation is lower
(r = 0.4) but each year has more effect on the ROM compared with
the plot on the right. The plot on the right shows a stronger
correlation (r = 0.8), but the age has little effect on the ROM.
Regression Analysis
Regression analyses are a set of statistical models used to estimate the
association between independent and dependent variables. For example,
using a regression model, we can define the association between age
(independent) and ROM (dependent). Most frequently used regression models
in medical research assume a linear relationship between the studied
variables, and, accordingly, are called linear regression models. These models
find the optimal line of fit to match the data to represent the relationship. This
line can be written as a mathematical function, which contains the coefficients
(β) of the independent variables in the model. Each β-coefficient represents
how much one unit of change in the independent variable affects the
dependent variable. For example, if we assessed the association between age
and ROM and β for age is −0.24, it means that each additional year of age
decreases the mean ROM by 0.24° (and 10 years by 2.4°) (Figure 16.10).
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FIGURE 16.10. Plots of two linear models. The plot on the left
shows association between age and range of motion (ROM) of
finger. The intercept is the ROM value when age = 0. The coefficient
β shows how much each year affects the average age, and the Rsquared value indicates how much of the variation in the ROM is
explained by the age. Here, most of the variation is related to other
factors than age. On the right, two groups (collagenase and surgery)
are compared with a linear model. The intercept (20.5°) is the mean
improvement in group 1 (collagenase), and β presents the betweengroup difference (11.7°). As on the left panel, R-squared presents
the amount variation in improvement explained by the treatment.
The choice of model depends on the outcome variable. When the outcome
is a continuous variable such as ROM, the model is called linear regression.
With a dichotomous outcome, such as success or adverse event, the model is
called binary logistic regression, and with multinomial dependent outcomes,
multinomial logistic regression. Ordinal regression is used when the outcome
of interest is ordinal, and Cox proportional hazards regression is used when
the outcome is time to an event, such as death or recurrence.
Regression analysis is useful when multiple potential confounders are
present or when we want to see the independent effect of a variable on the
outcome. Stratification is extremely tedious when there are multiple variables.
Multivariable regression analysis enables us to simultaneously examine the
relationship between multiple independent variables and one dependent
variable. In an observational study comparing two cohorts (surgery vs
collagenase), linear regression is useful because the outcome (PIPJ motion) is
a continuous variable and the model adjusts for the age (possible confounder).
The analysis will provide an age-adjusted difference (the mean difference if
age was held constant) in final ROM between surgery and collagenase
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injection. In a similar way, multivariate regression refers to models that include
multiple correlated outcomes (multiple dependent variables). However, not all
confounders are known and measured, and residual confounding is always
possible.
DIAGNOSTIC TEST PERFORMANCE
A large part of clinical medicine involves evaluating diagnostic tests. Imagine
that there is a new Artificial Intelligence (AI)-powered test that can identify
melanoma based on photographs taken of a nevus. A surgeon would want to
know how accurate this algorithm was and if it could reliably differentiate
melanoma from similar-looking benign nevus.
New tests are often compared against an established standard, such as
validated clinical criteria, radiographic findings, intraoperative findings, or
histopathological features. Table 16.5 shows the four possible scenarios with
this hypothetical test.
TABLE 16.5. DIAGNOSTIC TEST PERFORMANCE. THE CELLS OF
THE TABLE INDICATE THE HYPOTHETICAL RESULTS OF A NEW AI
TEST FOR MELANOMA, AND THESE ARE CONVENTIONALLY
LABELED A, B, C, D IN CLOCKWISE MANNER
Melanoma No Melanoma
Test positive 196 [a = true positive] 5 [b = false positive]
Test negative 4 [c = false negative] 95 [d = true negative]
The sensitivity of the test is the proportion of patients with positive test result
and who have the disease (true positive). Specificity is the proportion of
patients who do not have the disease who test negative (true negative).
10
These can be calculated as follows:
Sensitivity = a/(a + c)
Sensitivity = d/(b + d)
In the example shown in Table 16.5, the new AI test sensitivity is 98% and
the specificity is 95%. A highly sensitive test has a high true-positive rate and a
negative result rules out disease—“sensitive rules out” (SNout). A specific test
has a low false-positive rate, and a positive test “rules in” disease (SPin).
2
A likelihood ratio (LR) is a useful measure to indicate the diagnostic
capability of a test. It conveys information on how much a positive or negative
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test alters the probability of the condition being present. For example, in the
previous example, the positive LR indicates how much a positive test affects
the pretest likelihood of melanoma and negative LR indicates how much a
negative test decreases the likelihood of melanoma.
Sensitivity and specificity indicate the probability of positive or negative test
result given the disease status of the patient. However, the physician does not
actually know the true disease status of the patient and is more interested in
the probability of the patient having the disease given a positive test. This is
the positive predictive value (PPV). Similarly, the negative predictive value
(NPV) is the probability of the patient not having the condition given a negative
test. Unlike sensitivity and specificity, PPV and NPV depend on the prevalence
of the disease in the population. In our hypothetical example, the PPV of the AI
diagnostic test would be higher in a Caucasian population where the
prevalence of melanoma is higher but lower in an African population where
melanoma is much less common. LR, PPV, and NPV are properties of the test
that are important from a public health perspective. Their application is context
dependent. No test is perfect and policy makers often need to determine
diagnostic thresholds based on the implications of a false-positive or falsenegative result. A false-negative test for benign disease is probably not a
serious problem, but missing a diagnosis of early melanoma has far graver
consequences.
In some cases, a test yields a continuous result, and we need to determine
the optimal cutoff value to categorize individuals. For example, assume that
the algorithm described above yields a continuous index that tends to be
higher in people with melanoma. By using a receiver operating characteristic
(ROC) curve, we can identify the optimal cutoff value that most accurately
discriminates melanomas from benign lesions. The association between
sensitivity and specificity can be graphically displayed (Figure 16.11), and the
area under the curve (AUC) indicates the diagnostic performance of the test.
10
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FIGURE 16.11. An ROC curve presenting a hypothetical ROC
curve for index meant to detect melanoma. The optimal cutoff value
to classify patients as having or not having melanoma is an index of
8.59 points with sensitivity of 0.829 and specificity of 0.782. The area
under curve (AUC) is 0.838 suggesting that the test performs better
than chance (AUC 0.5 corresponds with chance and 1.0 with perfect
discrimination).
META-ANALYSIS
A meta-analysis yields a quantitative synthesis of the existing evidence by
combining data from several studies that investigate the same research
question, such as the effect of collagenase in patients with Dupuytren
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contracture. A meta-analysis is typically performed together with a systematic
review. Although meta-analytical methods can be applied to various types of
quantitative studies (such as observational studies and predictive models),
clinicians are more interested in meta-analyses looking at interventions, and
reliable information for these may be obtained from RCTs. Before performing
the actual analysis, the authors need to define the research question, identify
all relevant studies, extract the data, assess whether studies are clinically
similar (homogenous), and assess the risk of bias of the included studies.
Meta-analysis typically involves assigning weights to studies based on the
amount of information each study contributes (inverse variance weighing). The
larger the study, the smaller the random variation, thus giving a study more
weight. It is also possible to use other kinds of weighting, but inverse variance
is the most employed technique.
Heterogeneity
Meta-analysis first calculates treatment effect and its standard error for each
study (Figure 16.12). The effects are pooled using either fixed-effect or
random-effect models. A fixed-effect model assumes that the studies are
similar (no heterogeneity) and that they measure the same (fixed) underlying
effect of intervention and the difference between the studies is purely because
of sampling variation. Conversely, a random-effect (RE) meta-analysis
assumes that there is heterogeneity between the included studies and the
effects in the individual studies come from a distribution of various effects.
Heterogeneity is typically assessed with I2 statistics. The I2 value reflects the
percentage of variation that is because of heterogeneity rather than random
variation. Heterogeneity can be also assessed with Cochrane Q-test, chisquared test, or Tau.11 If a large amount of unexplained heterogeneity is
present in a meta-analysis, the average effect may not represent a useful
estimate of the treatment effect, especially if the underlying reasons for the
heterogeneity are unclear. Other factors that are important to consider when
interpreting results of a meta-analysis are the risk of bias of the included
studies, whether studies used surrogate or direct measurement, whether effect
estimate is imprecise or not, and the presence of publication bias.12 The
evidence of publication bias is assessed indirectly using a funnel plot, which
may show asymmetry indicating that negative studies have been left
unpublished.
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FIGURE 16.12 A hypothetical forest plot showing meta-analysis of
six studies. All except study 6 suggest that the effect is clinically
unimportant (under minimal clinically important value). This study
alone causes considerable statistical heterogeneity (I2-value is 81%),
and the reasons for this should be explored. If study 6 had a high
risk of bias (eg, lack of blinding), then we might suspect that the
estimate is biased and the other studies reflect the effect of the
studied treatment better.
QUESTIONS
TABLE 16.6.
De Quervain
tenosynovitis (+)
De Quervain
tenosynovitis (−)
Pregnant in the last
12 mo
40 60
Not pregnant in the
last 12 mo
15 135
1. Several large hand surgery units recently concluded a multicenter
prospective study evaluating the relationship between developing de
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