Skip Navigation
Skip to contents

Restor Dent Endod : Restorative Dentistry & Endodontics

OPEN ACCESS

Articles

Page Path
HOME > Restor Dent Endod > Volume 44(3); 2019 > Article
Open Lecture on Statistics Statistical notes for clinical researchers: the independent samples t-test
Hae-Young Kimorcid
Restor Dent Endod 2019;44(3):e26.
DOI: https://doi.org/10.5395/rde.2019.44.e26
Published online: July 17, 2019

Department of Health Policy and Management, College of Health Science, andDepartment of Public Health Science, Graduate School, Korea University, Seoul,Korea.

Correspondence to Hae-Young Kim, DDS, PhD. Professor, Department of Health Policy and Management, Korea University College of Health Science, and Department of Public Health Science, Korea University Graduate School, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Korea. kimhaey@korea.ac.kr

Copyright © 2019. The Korean Academy of Conservative Dentistry

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

  • 891 Views
  • 40 Download
  • 33 Crossref
prev next
The t-test is frequently used in comparing 2 group means. The compared groups may be independent to each other such as men and women. Otherwise, compared data are correlated in a case such as comparison of blood pressure levels from the same person before and after medication (Figure 1). In this section we will focus on independent t-test only. There are 2 kinds of independent t-test depending on whether 2 group variances can be assumed equal or not. The t-test is based on the inference using t-distribution.
rde-44-e26-g001.jpg Figure 1 

Types of 2-sample t-test.

Download Figure Download Figure
The t-distribution was invented in 1908 by William Sealy Gosset, who was working for the Guinness brewery in Dublin, Ireland. As the Guinness brewery did not permit their employee's publishing the research results related to their work, Gosset published his findings by a pseudonym, “Student.” Therefore, the distribution he suggested was called as Student's t-distribution. The t-distribution is a distribution similar to the standard normal distribution, z-distribution, but has lower peak and higher tail compared to it (Figure 2).
rde-44-e26-g002.jpg Figure 2 

The t-distribution with various degrees of freedom (df) compared to z-distribution.

Download Figure Download Figure
According to the sampling theory, when samples are drawn from a normal-distributed population, the distribution of sample means is expected to be a normal distribution. When we know the variance of population, σ2, we can define the distribution of sample means as a normal distribution and adopt z-distribution in statistical inference. However, in reality, we generally never know σ2, we use sample variance, s2, instead. Although the s2 is the best estimator for σ2, the degree of accuracy of s2 depends on the sample size. When the sample size is large enough (e.g., n = 300), we expect that the sample variance would be very similar to the population variance. However, when sample size is small, such as n = 10, we could guess that the accuracy of sample variance may be not that high. The t-distribution reflects this difference of uncertainty according to sample size. Therefore the shape of t-distribution changes by the degree of freedom (df), which is sample size minus one (n − 1) when one sample mean is tested.
The t-distribution appears to be a family of distribution of which shape varies according to its df (Figure 2). When df is smaller, the t-distribution has lower peak and higher tail compared to those with higher df. The shape of t-distribution approaches to z-distribution as df increases. When df gets large enough, e.g., n = 300, t-distribution is almost identical with z-distribution. For the inferences of means using small samples, it is necessary to apply t-distribution, while similar inference can be obtain by either t-distribution or z-distribution for a case with a large sample. For inference of 2 means, we generally use t-test based on t-distribution regardless of the sizes of sample because it is always safe, not only for a test with small df but also for that with large df.
To adopt z- or t-distribution for inference using small samples, a basic assumption is that the distribution of population is not significantly different from normal distribution. As seen in Appendix 1, the normality assumption needs to be tested in advance. If normality assumption cannot be met and we have a small sample (n < 25), then we are not permitted to use ‘parametric’ t-test. Instead, a non-parametric analysis such as Mann-Whitney U test should be selected.
For comparison of 2 independent group means, we can use a z-statistic to test the hypothesis of equal population means only if we know the population variances of 2 groups, σ21rde-44-e26-i002.jpg and σ22rde-44-e26-i003.jpg, as follows;
(Eq. 1)
Z=X1X2σ21n1+σ22n2
where 1 and 2, σ21rde-44-e26-i002.jpg and σ22rde-44-e26-i003.jpg, and n1 and n2 are sample means, population variances, and the sizes of 2 groups.
Again, as we never know the population variances, we need to use sample variances as their estimates. There are 2 methods whether 2 population variances could be assumed equal or not. Under assumption of equal variances, the t-test devised by Gosset in 1908, Student's t-test, can be applied. The other version is Welch's t-test introduced in 1947, for the cases where the assumption of equal variances cannot be accepted because quite a big difference is observed between 2 sample variances.
1. Student's t-test
In Student's t-test, the population variances are assumed equal. Therefore, we need only one common variance estimate for 2 groups. The common variance estimate is calculated as a pooled variance, a weighted average of 2 sample variances as follows;
(Eq. 2)
s2p=(n11)(n11)+(n21)s21+(n21)(n11)+(n21)s22
where s21rde-44-e26-i004.jpg and s22rde-44-e26-i005.jpg are sample variances.
The resulting t-test statistic is a form that both the population variances, σ21rde-44-e26-i002.jpg and σ21rde-44-e26-i003.jpg, are exchanged with a common variance estimate, s2prde-44-e26-i006.jpg. The df is given as n1 + n2 − 2 for the t-test statistic.
(Eq. 3)
t=X1X2s2pn1+s2pn2=X1X2sp1n1+1n2~t(n1+n22)
In Appendix 1, ‘(E-1) Leven's test for equality of variances’ shows that the null hypothesis of equal variances was accepted by the high p value, 0.334 (under heading of Sig.). In ‘(E-2) t-test for equality of means t-values’, the upper line shows the result of Student's t-test. The t-value and df are shown −3.357 and 18. We can get the same figures using the formulas Eq. 2 and Eq. 3, and descriptive statistics in Table 1, as follows.
Table 1

Descriptive statistics and result of the Student's t-test

Group No. Mean Standard deviation p value
1 10 10.28 0.5978 0.004
2 10 11.08 0.4590
Download Table Download Table
s2p=(101)×0.59782+(101)×0.45902(101)+(101)=5.112418=0.2840=(0.5329)2
t=X1X2sp1n1+1n2=10.2811.080.532919+19=0.800.2512=3.18
df = n1 + n2 − 2 = 10 + 10 − 2 = 18
The result of calculation is a little different from that by SPSS (IBM Corp., Armonk, NY, USA) of Appendix 1, maybe because of rounding errors.
2. Welch's t-test
Actually there are a lot of cases where the equal variance cannot be assumed. Even if it is unlikely to assume equal variances, we still compare 2 independent group means by performing the Welch's t-test. Welch's t-test is more reliable when the 2 samples have unequal variances and/or unequal sample sizes. We need to maintain the assumption of normality.
Because the population variances are not equal, we have to estimate them separately by 2 sample variances, s21rde-44-e26-i004.jpg and s22rde-44-e26-i005.jpg. As the result, the form of t-test statistic is given as follows;
(Eq. 4)
t=X1X2s21n1+s22n2~tν
where ν is Satterthwaite degrees of freedom.
(Eq. 5)
ν=s21n1+s22n2(s21n1)2/(n11)+(s22n2)2/(n21)
In Appendix 1, ‘(E-1) Leven's test for equality of variances’ shows an equal variance can be successfully assumed (p = 0.334). Therefore, the Welch's t-test is inappropriate for this data. Only for the purpose of exercise, we can try to interpret the results of Welch's t-test shown in the lower line in ‘(E-2) t-test for equality of means t-values’. The t-value and df are shown as −3.357 and 16.875.
t=X1X2s21n1+s22n2=10.2811.080.5978210+0.4590210=0.800.2383=3.357
υ=(s21n1+s22n2)2(s21n1)2/(n11)+(s22n2)2/(n21)=(0.5978210+0.4590210)2(0.5978210)2101+(0.4590210)2101=0.0567920.0001419+0.0000493=.0032250.000191216.87
We've confirmed nearly same results by calculation using the formula and by SPSS software.
The t-test is one of frequently used analysis methods for comparing 2 group means. However, sometimes we forget the underlying assumptions such as normality assumption or miss the meaning of equal variance assumption. Especially when we have a small sample, we need to check normality assumption first and make a decision between the parametric t-test and the nonparametric Mann-Whitney U test. Also, we need to assess the assumption of equal variances and select either Student's t-test or Welch's t-test.
Appendix 1

Procedure of t-test analysis using IBM SPSS

The procedure of t-test analysis using IBM SPSS Statistics for Windows Version 23.0 (IBM Corp., Armonk, NY, USA) is as follows.
rde-44-e26-a001.jpg

Tables & Figures

rde-44-e26-g001.jpg Figure 1 

Types of 2-sample t-test.

Download Figure Download Figure
rde-44-e26-g002.jpg Figure 2 

The t-distribution with various degrees of freedom (df) compared to z-distribution.

Download Figure Download Figure
Table 1

Descriptive statistics and result of the Student's t-test

Group No. Mean Standard deviation p value
1 10 10.28 0.5978 0.004
2 10 11.08 0.4590
Download Table Download Table

REFERENCES

    Citations

    Citations to this article as recorded by  
    • Examining Gender Differences in Aggression as a Predictor of Anxiety, Depression, and Suicide in a Cross‐Sectional French Sample
      Sylvia Martin
      Health Science Reports.2025;[Epub]     CrossRef
    • Structural and topological analysis of thiosemicarbazone-based metal complexes: computational and experimental study of bacterial biofilm inhibition and antioxidant activity
      Doaa S. El‑Sayed, Shaymaa S. Hassan, Liblab S. Jassim, Ali Abdullah Issa, Firas AL-Oqaili, Mustafa k. Albayaty, Buthenia A. Hasoon, Majid S. Jabir, Khetam H. Rasool, Hemmat A. Elbadawy
      BMC Chemistry.2025;[Epub]     CrossRef
    • Comparison of the effects of video conference and video-based home exercise on physical performance and body composition in older adult individuals
      Özgün Elmas, Mustafa Cemali, Ayşe Livanelioğlu
      Medicine.2024; 103(44): e40329.     CrossRef
    • Deep Residual Learning-Based Classification with Identification of Incorrect Predictions and Quantification of Cellularity and Nuclear Morphological Features in Digital Pathological Images of Common Astrocytic Tumors
      Yen-Chang Chen, Shinn-Zong Lin, Jia-Ru Wu, Wei-Hsiang Yu, Horng-Jyh Harn, Wen-Chiuan Tsai, Ching-Ann Liu, Ken-Leiang Kuo, Chao-Yuan Yeh, Sheng-Tzung Tsai
      Cancers.2024; 16(13): 2449.     CrossRef
    • The Impact of Stress First Aid on Perceived Stress Levels of New Graduate Nurses
      John R. Balcuk
      Nursing Economic$.2024; 42(4): 191.     CrossRef
    • CloudBrain-MRS: An intelligent cloud computing platform for in vivo magnetic resonance spectroscopy preprocessing, quantification, and analysis
      Xiaodie Chen, Jiayu Li, Dicheng Chen, Yirong Zhou, Zhangren Tu, Meijin Lin, Taishan Kang, Jianzhong Lin, Tao Gong, Liuhong Zhu, Jianjun Zhou, Ou-yang Lin, Jiefeng Guo, Jiyang Dong, Di Guo, Xiaobo Qu
      Journal of Magnetic Resonance.2024; 358: 107601.     CrossRef
    • Bactericidal activities and biochemical features of 16 antimicrobial peptides against bovine-mastitis causative pathogens
      Hye-sun Cho, Dohun Kim, Hyoim Jeon, Prathap Somasundaram, Nagasundarapandian Soundrarajan, Chankyu Park
      Veterinary Research.2024;[Epub]     CrossRef
    • Expressions of Interleukin-4 and Interleukin-5 in Nodular Prurigo and Pruritic Papular Lesions
      Ayu Wikan Sayekti, Ann Kautsaria Putri, Dwi Retno Adi Winarni, Satiti Retno Pudjiati
      Folia Medica Indonesiana.2024; 60(1): 47.     CrossRef
    • Challenges and barriers to e-leadership participation: Examining the perspectives of Malaysian secondary school teachers
      Cha Shi Ping, Lokman Mohd Tahir, Mohd Shafie Rosli, Noor Azean Atan, Mohd Fadzli Ali
      Education and Information Technologies.2024; 29(8): 10329.     CrossRef
    • One-Leg Standing Test with Eyes Open as a Screening Tool for Prefrailty in Community-Dwelling Older Japanese Women
      Zhenyue Liu, Shuji Sawada, Hisashi Naito, Shuichi Machida
      Healthcare.2024; 12(23): 2378.     CrossRef
    • Detection of cardiovascular disease cases using advanced tree-based machine learning algorithms
      Fariba Asadi, Reza Homayounfar, Yaser Mehrali, Chiara Masci, Samaneh Talebi, Farid Zayeri
      Scientific Reports.2024;[Epub]     CrossRef
    • Recent Advancements in Subcellular Proteomics: Growing Impact of Organellar Protein Niches on the Understanding of Cell Biology
      Vanya Bhushan, Aleksandra Nita-Lazar
      Journal of Proteome Research.2024; 23(8): 2700.     CrossRef
    • A comparison study: The use of digital and conventional impression techniques in dental hygiene education
      Raha K. Naderi, Tulsi J. Patel, Michelle A. Thompson
      Journal of Dental Education.2024; 88(5): 518.     CrossRef
    • An Evaluation of the Nunez Community College Quality Enhancement Program: Assessing the Impact of Embedded Conover Workplace Readiness® Modules on Students’ Work Readiness
      Sandra Aurand Bosch, Donna Mowery Rice
      Sage Open.2024;[Epub]     CrossRef
    • Integrating electric vehicles into hybrid microgrids: A stochastic approach to future-ready renewable energy solutions and management
      Aykut Fatih Güven
      Energy.2024; 303: 131968.     CrossRef
    • Comparison of postural assessment and awareness in individuals receiving posture training using the digital AI posture assessment and correction system
      Musa Çankaya, Fatma Nur Takı
      International Journal of Occupational Safety and Ergonomics.2024; 30(4): 1311.     CrossRef
    • Detection and Analysis of Fake News Users’ Communities in Social Media
      Abdelouahab Amira, Abdelouahid Derhab, Samir Hadjar, Mustapha Merazka, Md. Golam Rabiul Alam, Mohammad Mehedi Hassan
      IEEE Transactions on Computational Social Systems.2024; 11(4): 5050.     CrossRef
    • How do fish miss? Attack strategies of threespine stickleback capturing non-evasive prey
      Seth Shirazi, Timothy E. Higham
      Journal of Experimental Biology.2024;[Epub]     CrossRef
    • Effects of Occlusal Contact on Maxillary Alveolar Bone Morphology in Patients with and without Anterior Open Bite: A Cross-Sectional Study
      Chiyo Shimizu-Tomoda, Yuji Ishida, Aiko Ishizaki-Terauchi, Yukari Mizoguchi, Shuji Oishi, Takashi Ono
      Journal of Clinical Medicine.2024; 13(11): 3061.     CrossRef
    • Bioceramic modular tissue-engineered bone with rapid vascularization for large bone defects
      Siwei Luo, Zhen Wang, Jialin He, Geng Tang, Daizhu Yuan, Zhanyu Wu, Zihao Zou, Long Yang, Tao Lu, Chuan Ye
      Ceramics International.2024; 50(11): 18275.     CrossRef
    • Effect of Arogya Raksha Panchatantra (five lifestyle principles) on heart rate variability, menstrual symptoms, health-related quality of life, performance and self-efficacy in Young female adults with primary dysmenorrhea: protocol for an exploratory ran
      Karishma Silwal, Prakash Babu Kodali, Vakeel Khan, Hemanshu Sharma, Gulab Rai Tewani, Pradeep M. K. Nair
      CCRYN Indian Journal of Yoga & Naturopathy.2024; 1(1): 15.     CrossRef
    • Letter to the editor on: “Calcaneal positioning in equinus immobilization of the ankle joint: A comparison of common orthoses used in the treatment of acute Achilles tendon ruptures”
      Rahul Kumar, Diggaj Shrestha, Kunal Setia, Sunita Sharma
      Foot and Ankle Surgery.2023; 29(6): 498.     CrossRef
    • Irradiation damage reduces alloy corrosion rate via oxide space charge compensation effects
      Zefeng Yu, Elizabeth Kautz, Hongliang Zhang, Anton Schneider, Taeho Kim, Yongfeng Zhang, Sten Lambeets, Arun Devaraj, Adrien Couet
      Acta Materialia.2023; 253: 118956.     CrossRef
    • Predictors of Relevant Changes in Pain and Function for Adolescents With Idiopathic Scoliosis Following Surgery
      Samia Alamrani, Adrian Gardner, Alison B. Rushton, Deborah Falla, Nicola R. Heneghan
      Spine.2023; 48(16): 1166.     CrossRef
    • A multicenter feasibility study on implementing a brief mindful breathing exercise into regular university courses
      Annika C. Konrad, Veronika Engert, Reyk Albrecht, Christian Dobel, Nicola Döring, Jens Haueisen, Olga Klimecki, Mike Sandbothe, Philipp Kanske
      Scientific Reports.2023;[Epub]     CrossRef
    • Adult children of parents with mental illness: Family stigma and coping on sense of self
      Chynna Campbell, Pamela Patrick
      Child & Family Social Work.2023; 28(3): 622.     CrossRef
    • Veri Madenciliğine Dayalı Olarak Çalışanların Örgütsel Bağlılık Düzeyinin Belirlenmesi: İstanbul ve Kocaeli Örneği
      Nadir ERSEN, Timuçin BARDAK, Uğur Can USTA
      Bartın Orman Fakültesi Dergisi.2023; 25(3): 398.     CrossRef
    • Pooled rates and demographics of POTS following SARS-CoV-2 infection versus COVID-19 vaccination: Systematic review and meta-analysis
      Shin Jie Yong, Alice Halim, Shiliang Liu, Michael Halim, Ahmad A. Alshehri, Mohammed A. Alshahrani, Mohammed M. Alshahrani, Amal H. Alfaraj, Lamees M. Alburaiky, Faryal Khamis, Muzaheed, Bashayer M. AlShehail, Mubarak Alfaresi, Reyouf Al Azmi, Hawra Alba
      Autonomic Neuroscience.2023; 250: 103132.     CrossRef
    • Rapid suture-free repair of arterial bleeding: A novel approach with ultra-thin bioadhesive hydrogel membrane
      Siwei Luo, Long Yang, Qiang Zou, Daizhu Yuan, Shunen Xu, Yanchi Zhao, Xin Wu, Zhen Wang, Chuan Ye
      Chemical Engineering Journal.2023; 472: 144865.     CrossRef
    • The Relationship Between Obsessive-Compulsive Disorder and Gaming Disorder
      Nazir Hawi, Maya Samaha
      International Journal of Cyber Behavior, Psychology and Learning.2023; 13(1): 1.     CrossRef
    • Microbiological Quality and Presence of Foodborne Pathogens in Raw and Extruded Canine Diets and Canine Fecal Samples
      Doina Solís, Magaly Toro, Paola Navarrete, Patricio Faúndez, Angélica Reyes-Jara
      Frontiers in Veterinary Science.2022;[Epub]     CrossRef
    • Stapler Assisted Total Laryngectomy: A Prospective Randomized Clinical Study
      Omar Ahmed, Hesham Mustafa Abdel-Fattah, Hisham E. M. Elbadan
      Indian Journal of Otolaryngology and Head & Neck Surgery.2022; 74(S2): 2205.     CrossRef
    • Project VLOGI (Video Lectures on Giving Instructions): Effects on Learners’ Performance in Probability and Statistics
      Sherwin BATİLANTES
      International Journal of Educational Studies in Mathematics.2021; 8(4): 299.     CrossRef
    CanvasJS.com
    CanvasJS.com
    CanvasJS.com

    • ePub LinkePub Link
    • Cite
      CITE
      export Copy Download
      Close
      Download Citation
      Download a citation file in RIS format that can be imported by all major citation management software, including EndNote, ProCite, RefWorks, and Reference Manager.

      Format:
      • RIS — For EndNote, ProCite, RefWorks, and most other reference management software
      • BibTeX — For JabRef, BibDesk, and other BibTeX-specific software
      Include:
      • Citation for the content below
      Statistical notes for clinical researchers: the independent samples t-test
      Restor Dent Endod. 2019;44(3):e26  Published online July 17, 2019
      Close
    • XML DownloadXML Download
    Figure
    • 1
    Statistical notes for clinical researchers: the independent samples t-test
    Image Image
    Figure 1 Types of 2-sample t-test.
    Figure 2 The t-distribution with various degrees of freedom (df) compared to z-distribution.
    Statistical notes for clinical researchers: the independent samples t-test

    Descriptive statistics and result of the Student's t-test

    GroupNo.MeanStandard deviationp value
    11010.280.59780.004
    21011.080.4590
    Table 1 Descriptive statistics and result of the Student's t-test


    Restor Dent Endod : Restorative Dentistry & Endodontics
    Close layer
    TOP Mpgyi