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Negative relationship

In statistics, a negative relationship (also known as negative correlation or inverse relationship) between two variables occurs when an increase in one variable is associated with a decrease in the other, and vice versa.^[1] This contrasts with a positive relationship, where both variables move in the same direction, and a zero relationship, where no consistent pattern exists. Negative relationships are fundamental in data analysis, appearing in fields such as natural sciences, social sciences, economics, and finance, where they help model phenomena like trade-offs or opposing trends.^[2] The strength and direction of a negative relationship are typically quantified using the Pearson correlation coefficient (r), which ranges from -1 (perfect negative relationship) to 0 (no linear relationship), with values closer to -1 indicating stronger inverse associations.^[3] For example, in natural sciences, temperature and solubility of gases in water exhibit a negative relationship—as temperature rises, gas solubility decreases. In economics, there is often a negative correlation between unemployment rates and inflation (as per the Phillips curve), and in finance, stock prices may negatively correlate with bond yields during market shifts.^[4]^[5] Interpreting negative relationships requires caution to distinguish correlation from causation, as spurious associations can arise from confounding factors.^[6] Visualization tools like scatter plots reveal these patterns, with points trending downward from left to right, while regression analysis can model and predict based on them. Understanding negative relationships aids in hypothesis testing, risk assessment, and decision-making across disciplines, highlighting the interconnected yet oppositional nature of variables in real-world data.

Conceptual Foundation

Definition

In interpersonal psychology, a negative relationship refers to a social tie characterized by behaviors that are irritating, demanding, critical, or upsetting, which undermine emotional well-being and often involve perceptions of a partner "getting on one's nerves" or making excessive demands.^[7] This contrasts with positive relationships, which involve supportive or affectionate exchanges, and neutral or ambivalent ties, where adverse experiences do not predominate. Negative relationships emphasize harmful dynamics that erode trust, increase conflict, and foster distress, occurring in contexts such as marriages, family bonds, friendships, or workplace interactions.^[8] Unlike toxic or abusive dynamics, which involve overt harm, negative relationships focus on everyday irritations without presupposing severe maltreatment.^[9] Central to the concept is the recognition that social ties can have independent positive and negative dimensions, rather than being opposites on a single continuum. Negative aspects, such as criticism or intrusion, independently predict poorer outcomes like elevated negative affect and reduced well-being, distinct from the mere absence of positivity.^[10] For example, in close relationships, negativity may stem from unmet expectations or poor emotion regulation, while in distant ties, it arises from infrequent but tense interactions. The term "negative social exchanges" or "difficult ties" is often used interchangeably, highlighting the bidirectional nature where both parties may experience strain.^[11] This definition assumes familiarity with social networks as interconnected relationships that influence health, where negative ties represent liabilities within otherwise supportive systems. Historically, the study of negative relationships in psychology emerged in the late 20th century within social support research, building on earlier ideas about the dual nature of social interactions. Seminal work by Karen Rook in the 1980s introduced the concept of negative social exchanges as distinct stressors, separate from positive support, with her 1984 analysis showing their unique impact on distress.^[12] This built on broader social psychology traditions from the 1970s, such as stress and coping models by Lazarus and Folkman, which recognized interpersonal conflict as a source of chronic strain. By the 1990s, researchers like Fincham and Linfield formalized the independence of positive and negative relationship evaluations, influencing modern views on ambivalence in ties.^[8]

Mathematical Representation

While negative relationships in interpersonal psychology are primarily conceptualized qualitatively through self-reports and observational methods, they can be represented mathematically in models of social network analysis and stress appraisal. For instance, the impact of negative ties on well-being can be modeled using additive equations where overall relationship quality Q is the sum of positive P and negative N components, such that Q = P - wN, with w > 1 reflecting the greater weight of negativity (as per the "bad is stronger than good" principle).^[13] Here, N quantifies frequency or intensity of irritations, often scaled from 0 to 1 based on validated measures like the Network of Relationships Inventory, illustrating how negativity disproportionately erodes Q.^[7] In theoretical frameworks, such as the Strength and Vulnerability Integration (SAVI) model, negative exchanges are represented as stressors moderated by age and controllability: emotional reactivity R = f(N, A, C), where A is age (older adults show reduced R due to better regulation), and C is controllability (high C mitigates impact). This functional form highlights that unavoidable negativity amplifies distress more than avoidable instances.^[7] Similarly, Socioemotional Selectivity Theory (SST) posits time horizons influencing negativity perception, modeled as a decay function where negativity salience decreases with perceived limited time: S_N = N \cdot e^{-t/T}, with t as age-related time perception and T as lifespan horizon.^[14] Geometrically, social networks with negative ties can be visualized as signed graphs, where positive edges are +1 and negative -1, and balance theory assesses stability: a triangle of ties is balanced if the product of signs is positive, but negative ties introduce imbalance leading to conflict. This vector approach equates relational strain to the angle between tie vectors, with obtuse angles (>90°) indicating opposition and heightened stress.^[10] A "perfect" negative relationship, though rare, would correspond to a correlation of -1 in affect scales, where every interaction yields maximal distress without positive offset.

Measurement Techniques

Self-Report Scales

Self-report scales are primary tools for assessing negative relationships in interpersonal psychology, capturing individuals' perceptions of irritating, demanding, critical, or upsetting behaviors in social ties. These scales typically use Likert-type items to quantify the frequency or intensity of negative interactions, allowing differentiation from positive or ambivalent dynamics. Common scales focus on specific aspects like emotional distress or conflict, with responses often ranging from "not at all" to "extremely."^[15] The Social Relationships Index (SRI), developed by Uchino et al., measures both positivity and negativity in social networks. It includes a negativity subscale with three items, such as "How much has this person upset you when you needed understanding?" rated on a 6-point scale (1 = not at all, 6 = very much). This subscale has good internal consistency (Cronbach's α = 0.80) and test-retest reliability (r = 0.52–0.72 over 2 weeks). The SRI is suitable for evaluating close ties like spouses or friends and has been validated against other relationship quality measures, showing convergent validity (r = 0.50–0.56 with negativity aspects of the Quality of Relationships Inventory). It helps classify relationships as aversive or ambivalent, aiding research on health impacts.^[15] Another widely used tool is the Positive–Negative Relationship Quality (PN-RQ) scale, which treats positive and negative qualities as distinct dimensions. The negative subscale includes items like "frustrating" or "disappointing," rated on a 0–5 scale, with higher scores indicating greater negativity. Developed through three studies in 2016, it demonstrates strong reliability (Cronbach's α > 0.85 for subscales) and validity, correlating with marital satisfaction and conflict measures. The PN-RQ is applicable to various relationships, including romantic and familial, and avoids conflating absence of positivity with negativity.^[16] Limitations of self-report scales include potential response biases, such as social desirability, and subjectivity in perceptions. They are most effective when combined with multi-item assessments and validated in diverse populations, with thresholds for negativity often context-specific (e.g., mean scores above 3 indicating moderate negativity).^[17]

Advanced Assessment Methods

Advanced methods extend self-reports to capture nuanced negative dynamics, including network analysis and mixed-methods approaches for identifying antagonistic ties. These techniques enable prediction of relational outcomes and intervention needs, often incorporating qualitative insights alongside quantitative scores. The Types of Negative Ties (TNT) Scale, developed in 2023 using mixed methods, assesses specific negative tie types like worrying, manipulating, or annoying behaviors. It comprises 13 items (e.g., "This person got on your nerves" or "This person did not support you in case of need"), rated on a 7-point frequency scale (1 = never, 7 = always). Factor analysis revealed three dimensions (manipulating, unfaithful, worrying), with high reliability (KMO = 0.829) and validity supported by qualitative interviews (N = 444). This scale is particularly useful for social network studies, distinguishing ego- and alter-perceived negativity, and applies to broad ties beyond close relationships.^[18] Name-generator methods, reviewed systematically in 2024, identify antagonistic ties by prompting respondents to list network members eliciting negative responses (e.g., "Who upsets you most?"). These can be combined with rating scales for depth, offering ecological validity in community samples. Reliability varies by prompt specificity, but they enhance understanding of negativity prevalence (e.g., 10–20% of ties in adult networks as negative). Such methods support longitudinal tracking of relational strain and its health correlates.^[19] In research, data from these measures are analyzed using statistical techniques like regression to model negativity's effects on well-being, controlling for confounders. For instance, negative subscale scores predict outcomes via models where higher negativity (β < 0 for well-being) indicates strain. However, the core measurement remains perceptual assessment rather than inferential modeling. Validity is tested through associations with criteria like conflict frequency, ensuring scales capture true relational detriment.^[20]

Visualization and Interpretation

Scatter Plots and Graphs

Scatter plots provide a fundamental visual method for identifying negative relationships between two variables by plotting paired data points (x, y) on a two-dimensional Cartesian plane. In such plots, a negative relationship manifests as a cluster of points forming a pattern that slopes downward from the upper left to the lower right, where higher values of one variable correspond to lower values of the other.^[21] This visual cue aligns with the negative slope described in the mathematical representation of such relationships.^[22] To enhance clarity, a trend line—often a straight line fitted through the points—can be overlaid to emphasize the overall downward direction and approximate the strength of the association.^[23] Key identification features in scatter plots include the tightness of the point clustering along the negative slope, which indicates a stronger relationship, compared to more dispersed points that suggest weaker or noisier associations.^[24] For instance, points hugging a steep downward line imply that changes in one variable reliably predict opposite changes in the other, while scattered points across the plot reveal limited predictability.^[25] This distinction aids in quickly assessing the presence and reliability of negative patterns without delving into numerical computations. Beyond basic scatter plots, other graph types extend visualization for specific contexts involving negative relationships. Line graphs are particularly effective for time-series data, where one variable is plotted against time on the x-axis, revealing a downward trajectory for negative trends, or dual lines can illustrate inverse movements between two variables over time.^[26] Bubble plots build on scatter plots for multivariate analysis, incorporating a third variable through the size of each point (bubble) while maintaining the positional indication of a negative relationship between the primary pair.^[27] Larger or smaller bubbles can thus highlight additional dimensions, such as magnitude or category, within the downward-sloping pattern.^[25] Common software tools for generating these visualizations include the R programming language with its ggplot2 package, Python's matplotlib library for customizable plotting, and Microsoft Excel for straightforward chart creation in spreadsheet environments.^[28] These tools enable users to input data pairs, select scatter or line options, and adjust axes to reveal negative patterns efficiently.^[29]

Distinguishing Causation from Correlation

In the context of negative relationships, a key challenge arises when observed correlations are misinterpreted as causal links, leading to erroneous conclusions about how one variable inversely affects another. Correlation measures, such as Pearson's coefficient, quantify the strength and direction of association but cannot establish causality on their own, as they do not account for underlying mechanisms or external influences. This distinction is particularly critical for negative correlations, where an inverse pattern might suggest that increases in one variable reduce the other, yet such patterns often stem from coincidence or hidden factors rather than direct influence. Spurious correlations represent coincidental negative associations without any causal connection, frequently arising from mathematical artifacts or unrelated trends. A classic example occurs in ratio variables, such as the percentage of a restaurant tip relative to bill size, which exhibits a spurious negative correlation with bill size itself. This arises because tips tend to be more proportionally variable for smaller bills, creating an illusory inverse relationship that dissipates when absolute tip amounts are analyzed instead.^[30] Such artifacts highlight how data transformations can inadvertently produce misleading negative patterns, emphasizing the need to scrutinize the form of measurement. Confounding variables further complicate interpretation by introducing third factors that distort the apparent negative relationship between two variables of interest. For instance, an observed negative correlation between income and habitual physical exercise may appear to suggest that higher earnings discourage activity, but this is confounded by age: older individuals often have higher incomes yet reduced exercise due to physical limitations, masking any true associations within age groups.^[31] Controlling for such confounders through stratification or multivariate adjustment reveals the underlying dynamics, preventing overattribution of causality to the negative association. In time-series data exhibiting negative relationships, the Granger causality test provides a statistical framework to assess potential directional influence, determining if past values of one variable help predict the other beyond its own history. Developed for econometric analysis, it evaluates whether lags of an independent variable significantly improve forecasts of the dependent variable, offering evidence of precedence but not true causation, as it assumes stationarity and can be sensitive to model specification. This approach is useful for negative correlations in dynamic systems, such as economic indicators, but requires complementary methods to rule out reverse or bidirectional effects. To rigorously test causality in negative relationships, researchers prioritize designs that isolate effects, such as randomized controlled trials, which randomly assign interventions to minimize confounding and establish directional impact through experimental manipulation. Where randomization is infeasible, instrumental variables—external factors affecting the treatment but not the outcome directly—enable causal estimation by breaking confounding links, as formalized in econometric literature. These practices ensure that negative associations are not misconstrued as causal without robust evidence.

Examples and Applications

In the natural sciences, a prominent example of a negative relationship is observed between temperature and the solubility of gases in water. As temperature increases, the solubility of most gases, such as oxygen and carbon dioxide, decreases because the kinetic energy of gas molecules rises, favoring their escape from the liquid phase into the gas phase.^[32] This principle, rooted in thermodynamic equilibria, explains phenomena like the reduced oxygen availability in warmer aquatic environments, impacting marine life.^[32] Another key instance in natural sciences involves vaccination rates and the incidence of vaccine-preventable diseases, such as measles. Higher vaccination coverage inversely correlates with disease cases, as evidenced by global trends where first-dose measles immunization rates rose from 72% in 2000 to 86% in 2019, accompanying an initial decline in estimated cases from ~37 million to ~10 million by 2019; however, coverage fell to 83% by 2023, correlating with a resurgence to ~10.3 million estimated cases. Reported cases fluctuated, from ~853,000 in 2000 but rising to ~870,000 by 2019 amid outbreaks, before dropping in 2020 due to COVID-19 reporting disruptions. By 2025, declining coverage has reversed gains, with outbreaks in every WHO region and over 1,700 confirmed U.S. cases—the highest since elimination in 2000.^[33] This relationship underscores how increased immunization disrupts disease transmission chains, reducing outbreak severity.^[33]^[34]^[35]^[36] In the social sciences, education level exhibits a negative relationship with unemployment rates. In 2024, across OECD countries, individuals aged 25-34 with below upper secondary education faced an average unemployment rate of 12.5%, nearly double the 6.8% rate for those with tertiary education, reflecting how higher education enhances employability through skill acquisition.^[37] Similarly, greater income inequality is associated with reduced social mobility, where countries with higher inequality experience lower intergenerational earnings mobility, as persistent income gaps limit opportunities for advancement across generations.^[38] These examples illustrate both cross-sectional and longitudinal analyses of negative relationships. Cross-sectional studies provide static snapshots, such as comparing unemployment rates across education levels within a single year like 2024, revealing immediate disparities.^[37] In contrast, longitudinal approaches track changes over time, like the multi-decade trends in measles incidence alongside vaccination coverage from 2000 onward, highlighting evolving trends.^[33] Such distinctions aid in interpreting whether relationships are situational or persistent.

In Economics and Finance

In economics, negative relationships are central to understanding how interest rates influence investment spending. According to the IS curve in Keynesian macroeconomics, higher real interest rates increase the cost of borrowing, thereby reducing firms' planned investment in capital goods and shifting the aggregate demand curve leftward, which lowers equilibrium output.^[39] This inverse dynamic is derived from the spending balance condition where investment (I) decreases as the interest rate (r) rises, ensuring goods market equilibrium at lower income levels.^[40] Empirical models often incorporate this relationship to predict how monetary policy tightening dampens economic activity.^[41] In finance, negative relationships underpin asset diversification strategies within Modern Portfolio Theory, where combining assets like stocks and bonds—often exhibiting negative correlations—lowers overall portfolio volatility without sacrificing expected returns. Harry Markowitz's foundational work demonstrates that when the correlation coefficient (ρ) between asset returns is negative, the portfolio's standard deviation decreases more effectively than with positively correlated assets, enabling risk-averse investors to optimize the efficient frontier.^[42] For instance, during periods of equity market downturns, bond prices typically rise due to flight-to-safety effects, reducing the combined risk.^[43] This principle was starkly illustrated in the 2008 financial crisis, where declining housing prices led to a surge in mortgage defaults; as home values fell by nearly 30% from peak to trough, overall serious delinquency rates rose from ~1.4% in late 2006 to ~3.8% by mid-2008 (and over 25% for subprime), amplifying systemic losses.^[44]^[45]^[46] Hedging strategies further exploit negative relationships by pairing assets that move inversely, such as gold and equities, to mitigate downside risk. Gold's returns often show negative correlation with stock indices during market stress, acting as a safe-haven asset that appreciates when equities decline, thereby stabilizing portfolio value.^[47] This dynamic strengthens in crises, as evidenced by gold's inverse movement against equities during the 2008 downturn and subsequent events, allowing investors to offset losses through targeted allocations.^[48] The Capital Asset Pricing Model (CAPM) provides empirical evidence of negative relationships through the beta coefficient (β), where β < 0 indicates an asset's returns move inversely to the market portfolio, implying lower systematic risk and potentially negative expected premiums in equilibrium.^[49] Developed by William Sharpe, CAPM posits that such assets hedge market risk, as their covariance with the market return is negative, leading to reduced portfolio variance when included.^[50] Rare in practice but observable in defensive sectors like utilities during recessions, negative betas underscore opportunities for inverse positioning in asset allocation.^[51]

Comparison to Positive and Zero Relationships

In statistics, a negative relationship between two variables is characterized by an inverse directional association, where an increase in one variable corresponds to a decrease in the other, as indicated by a Pearson correlation coefficient r < 0.^[52] This contrasts with a positive relationship, where both variables tend to move in the same direction—rising or falling together—yielding r > 0.^[53] The sign of r thus reveals the orientation of the linear association, with negative values highlighting oppositional patterns essential for certain analytical contexts.^[54] The Pearson correlation coefficient quantifies these relationships through the formula:

r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2 \sum_{i=1}^{n} (y_i - \bar{y})^2}}

Here, a positive numerator \sum (x_i - \bar{x})(y_i - \bar{y}) > 0 signifies a positive relationship, reflecting aligned deviations from the means, while a negative numerator indicates the inverse for negative relationships.^[55] A zero relationship, by comparison, arises when r \approx 0, implying no discernible linear association and producing a random scatter of data points around the line of best fit.^[56] To assess zero relationships, statistical tests evaluate the null hypothesis \rho = 0; for continuous variables, this often involves a t-test on the correlation coefficient, whereas for categorical data indicating no association, the chi-square test of independence is applied.^[55]^[57] Directionally, negative relationships facilitate risk mitigation strategies, such as in portfolio diversification under modern portfolio theory, where combining assets with r < 0 lowers overall variance without sacrificing expected returns.^[58] Positive relationships, conversely, support growth-oriented models, like those in economic forecasting where co-increasing variables amplify predictive synergies, though they may heighten volatility if overly aligned.^[53] Thresholds for interpreting r provide a structured comparison across relationship types, focusing on both direction and strength (where strength is assessed by |r|): | Relationship Type | r Range | Strength Interpretation (by |r|) | Key Characteristics | |-------------------|------------------------|---------------------------------------|---------------------| | Positive | $0 < [r](/page/R) \leq 1 | 0–0.3: Weak; 0.3–0.7: Moderate; ≥0.7: Strong | Variables move together; useful for synergy models.^[53] | | Zero | [r](/page/R) \approx 0 | None (no linear link) | Random scatter; independence likely.^[56] | | Negative | -1 \leq [r](/page/R) < 0 | 0–0.3: Weak; 0.3–0.7: Moderate; ≥0.7: Strong | Variables move oppositely; aids risk offset.^[53] |

Common Misconceptions

One prevalent misconception about negative relationships is that a negative correlation necessarily implies causation, such as assuming that an increase in one variable directly causes a decrease in another without considering alternative explanations.^[59] For instance, a negative association between prices and demand might be misinterpreted as prices causing reduced demand, overlooking potential confounders like seasonal supply variations or external market forces.^[60] This error stems from the broader pitfall that correlation, whether positive or negative, does not establish causality, as third variables or reverse directionality can create spurious inverse links.^[61] Another common misunderstanding is that all negative relationships are linear, leading analysts to overlook non-linear inverse patterns where the strength of the negative association varies across variable ranges.^[62] Pearson's correlation coefficient, for example, measures only linear associations and may yield a low or zero value for curved negative relationships, such as diminishing returns in economic models where initial increases in input yield strong inverse outputs but later taper off.^[63] In reality, non-linear negatives can exist in monotonic decreasing forms, and assuming linearity risks underdetecting meaningful inverses, as demonstrated in datasets with exponential decay patterns.^[59] Historically, pre-1950s statistical practice overemphasized parametric methods like Pearson's coefficient for negative relationships, often ignoring non-parametric alternatives that better handle non-normal data or outliers common in inverse scenarios.^[64] Developed in the late 19th century, Pearson's approach dominated until the mid-20th century, when non-parametric rank-based methods, such as Spearman's rho introduced in 1904, gained traction for capturing monotonic negatives without linearity assumptions; their widespread adoption accelerated post-1940s with contributions like Wilcoxon’s tests in 1945.^[65] In modern contexts, big data amplifies these issues by detecting weak negative correlations as statistically significant due to large sample sizes, yet it heightens the need to verify against confounders like measurement errors or aggregation biases that can artifactually invert relationships.^[66] To address these misconceptions, researchers should routinely check for confounding variables that might distort negative associations and employ multiple analytical methods, including non-parametric tests and graphical inspections, to confirm the nature and validity of inverse relationships.^[56]

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Cramer's V varies between 0 and 1 without any negative values. Similar to Pearson's r, a value close to 0 means no association. However, a value bigger than ...
[53]
Pearson Correlation Coefficient (r) | Guide & Examples - Scribbr
May 13, 2022 · “Absolute” means that if the t value is negative you should ignore the minus sign. Example: Comparing the t value to the critical value of t (t*) ...
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Interpreting Correlation Coefficients - Statistics By Jim
A coefficient of zero represents no linear relationship. As one variable increases, there is no tendency in the other variable to either increase or decrease.
[55]
Chi-Square (Χ²) Tests | Types, Formula & Examples - Scribbr
May 23, 2022 · A Pearson's chi-square test is a statistical test for categorical data. It is used to determine whether your data are significantly different from what you ...Chi-square table · Chi-Square Goodness of Fit Test · Of Independence
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[PDF] 5 – MODERN PORTFOLIO THEORY
perfectly correlated (ρ < 1) there are benefits of diversification. With perfect negative correlation (ρ = -1) the benefits are the greatest. For that ...
[57]
What Everyone Should Know about Statistical Correlation
Or the correlation can be negative: The increase in the value of one variable may be followed by the decrease in the value of the other.This Article From Issue · January-February 2015 · Page 26
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Correlation vs. Causation | Difference, Designs & Examples - Scribbr
Jul 12, 2021 · Mistaking correlation for causation is a common error and can lead to false cause fallacy. Why doesn't correlation imply causation?
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How to Distinguish Correlation from Causation in Orthopaedic ... - NIH
Correlations in observational studies are commonly misinterpreted as causation. Although correlation is necessary to establish a causal relationship between two ...
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Common pitfalls in statistical analysis: The use of correlation ... - NIH
By contrast, a “negative” correlation [Figure 1d-f] exists when increasing values of one variable are associated with a decrease in the values of the other.
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Myths About Linear and Monotonic Associations: Pearson's r ...
After defining linear and monotonic associations, we will demonstrate that these opinions are incorrect. Pearson's correlation coefficient should not be ruled ...Missing: misconception | Show results with:misconception
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Nonparametrics: The Early Years-Impressions and Recollections
Apr 28, 1983 · The Spearman rank correlation coefficient offered a striking example of the power of the theorem. Hotel- ling and Pabst (1936) had provided a ...Missing: pre- | Show results with:pre-
[63]
How Frank Wilcoxon helped statisticians walk the non-parametric path
Dec 7, 2015 · The term 'non-parametric' was first coined by Wolfowitz1 in 1942. Many non-parametric methods are based not on the actual magnitude of the ...
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Big Data and Large Sample Size: A Cautionary Note on the ...
We consider a variety of biases that are likely in the era of big data, including sampling error, measurement error, multiple comparisons errors, aggregation ...